An extension of ABC-theorem1

Vol. 1, No. 2, 2005 ISSN 1556-6706 SCIENTIA MAGNA Northwest University Xi’an, Shaanxi, P. R. China Vol. 1, No. 2, 2005 ISSN 1556-6706 SCIENTIA MAGNA Edited by Department of Mathematics Northwest University Xi’an, Shaanxi, P.R.China Scientia Magna is published annually in 200-300 pages per volume and 1,000 copies. It is also available in microfilm format and can be ordered (online too) from: Books on Demand ProQuest Information & Learning 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Michigan 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) URL: http://wwwlib.umi.com/bod/ Scientia Magna is a referred journal: reviewed, indexed, cited by the following journals: "Referativnyi Zhurnal" "Zentralblatt Für Mathematik" (Germany), and "Matematika" (Academia Nauk, Russia), "Mathematical Reviews" (USA), "Computing Review" (USA), Institute for Scientific Information (PA, USA), "Library of Congress Subject Headings" (USA). Price: US$ 69.95 Information for Authors Papers in electronic form are accepted. They can be e-mailed in Microsoft Word XP (or lower), WordPerfect 7.0 (or lower), LaTeX and PDF 6.0 or lower. The submitted manuscripts may be in the format of remarks, conjectures, solved/unsolved or open new proposed problems, notes, articles, miscellaneous, etc. They must be original work and camera ready [typewritten/computerized, format: 8.5 x 11 inches (21,6 x 28 cm)]. They are not returned, hence we advise the authors to keep a copy. The title of the paper should be writing with capital letters. The author's name has to apply in the middle of the line, near the title. References should be mentioned in the text by a number in square brackets and should be listed alphabetically. Current address followed by e-mail address should apply at the end of the paper, after the references. The paper should have at the beginning an abstract, followed by the keywords. All manuscripts are subject to anonymous review by three independent reviewers. Every letter will be answered. Each author will receive a free copy of the journal. Contributing to Scientia Magna Authors of papers in science (mathematics, physics, philosophy, psychology, sociology, linguistics) should submit manuscripts to the main editor: Prof. Zhang Wenpeng, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China. E-mail: wpzhang@nwu.edu.cn and another copy to: L. Cuciuc, 200 College Road, UNM-Gallup, NM 87301, U.S.A. E-mail: research@gallup.unm.edu. Associate Editors Dr. W. B. Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, IIT Madras, Chennai - 600 036, Tamil Nadu, India. Dr. Larissa Borissova and Dmitri Rabounski, Sirenevi boulevard 69-1-65, Moscow 105484, Russia. Dr. Liu Huaning, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China. E-mail: hnliu@nwu.edu.cn Prof. Yi Yuan, Research Center for Basic Science, Xi’an Jiaotong University, Xi’an, Shaanxi, P.R.China. E-mail: yuanyi@mail.xjtu.edu.cn Dr. Xu Zhefeng, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China. E-mail: zfxu@nwu.edu.cn Dr. Zhang Tianping, College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi, P.R.China. E-mail: tpzhang@snnu.edu.cn Contents V. Mladen and T. Krassimir : Remarks on some of the Smarandache’s problem. Part 2 1 W. Kandasamy : Smarandache groupoids 27 L. Ding : On the primitive numbers of power P and its mean value properties 36 D. Torres and V. Teca : Consecutive, reversed, mirror, and symmetric Smarandache sequence of triangular numbers 39 D. Ren : On the square-free number sequence 46 T. Ramaraj and N. Kannappa : On finite Smarandache near-rings 49 X. Kang : Some interesting properties of the Smarandache function 52 L. Mao : On Automorphism Groups of Maps, Surfaces and Smarandache Geometries 55 L. Ding : On the mean value of Smarandache ceil function 74 M. Le : An equation concerning the Smarandache function 78 M. Bayat, H. Teimoori and M. Hassani : An extension of ABC-theorem 81 J. Ma : An equation involving the Smarandache function 89 C. Chen : Inequalities for the polygamma functions with application 91 W. Vasantha and M. Chetry : On the number of Smarandache zero-divisors and Smarandache weak zero-divisors in loop rings 96 M. Le : The function equation S(n) = Z(n) 109 Z. Li : On the Smarandache Pseudo-number Sequences 111 D. Mehendale : The classical Smarandache function and a formula for twin primes 114 J. Sandor : On completely f -perfect numbers 116 L. Mao : Parallel bundles in planar map geometries 120 H. Yang and R. Fu : On the asymptotic properties of odd sieve sequence 134 A. Muktibodh : Smarandache quasigroup rings 139 H. Shen : Two asymptotic formulae on the k + 1-power free numbers 145 Y. Ji : An equation involving the Smarandache ceil function 149 J. Young : Smarandache BCC-algebras 152 M. Yang and Q. Yang : On the asymptotic property for Smarandache additive factorial complements 159 i J. Sandor : The Smarandache minimum and maximum functions 162 R. Pinch : Some properties of the Pseudo-Smarandache function 167 Y. Yi : An equation involving the Euler function and Smarandache function 173 J. Earls : Recursive Palindromic Smarandache Values 176 W. Vasantha and M. Chetry Smarandache Idempotents in finite ring Zn and in Group Ring Zn G 179 W. Vasantha and M. Chetry Smarandache Idempotents in Loop Rings Zt Ln (m) of the Loops Ln (m) 188 ii Scientia Magna Vol. 1 (2005), No. 2, 1-26 Remarks on some of the Smarandache’s problem. Part 2 Mladen V. Vassilev† , Missana and Krassimir T. Atanassov‡ † 5,V.Hugo Str., Sofia-1124, Bulgaria e-mail:missana@abv.bg ‡ CLBME-Bulg. Academy of Sci., P.O.Box 12, Sofia-1113, Bulgaria, e-mail:krat@bas.bg To Dr. Florentin Smarandache for his 50th birthday 0. In 1999, the second author of this remarks published a book over 30 of Smarandache’s problems in area of elementary number theory (see [1, 2]). After this, we worked over new 20 problems that we collected in our book [28]. These books contain Smarandache’s problems, described in [10, 16]. The present paper contains some of the results from [28]. In [16] Florentin Smarandache formulated 105 unsolved problems, while in [10] C.Dumitresu and V. Seleacu formulated 140 unsolved problems of his. The second book contains almost all the problems from [16], but now each problem has unique number and by this reason in [1, 28] and here the authors use the numeration of the problems from [10]. In the text below the following notations are used. N - the set of all natural numbers (i.e., the set of all positive integers); [x] - ”floor function” (or also so called ”bracket function”) - the greatest integer which is not greater than the real non-negative number x; ζ - Riemann’s Zeta-function; Γ - Euler’s Gamma-function; π - the prime counting function, i.e., π(n) denotes the number of prime p such that p ≤ n; ]x[ - the largest natural number strongly smaller than the real (positive) number x; dxe - the inferior integer part of x, i.e, the smallest integer greater than or equal to x. For an arbitrary increasing sequence of natural number C ≡ {cn }∞ n=1 we denote by πC (n) the number of terms of C, which are not greater than n. When n < c1 we put πC (n) = 0. 1. The results in this section are taken from [8]. The second problem from [10] (see also 16-th problem from [16]) is the following: Smarandache circular sequence: 1 , 12, 21, 123, 231, 312, 1234, 2341, 3412, 4123, |{z} | {z } | {z } | {z } 1 2 3 4 12345, 23451, 34512, 45123, 51234, | {z } 5 2 Mladen V. Vassilev, Missana and Krassimir T. Atanassov 123456, 234561, 345612, 456123, 561234, 612345, · · · | {z } 6 Let f (n) be the n-th member of the above sequence. We shall prove the following Theorem 1.1. For each natural number n: f (n) = s(s + 1) . . . k12 . . . (s − 1), where √ k ≡ k(n) =] 8n + 1 − 1 [ 2 and s ≡ s(n) = n − k(k + 1) . 2 2. The results in this section are taken from [25]. The eight problem from [10] (see also 16-th problem from [16]) is the following: Smarandache mobile periodicals (I): ...0 0 0 0 0 0 1 0 0 0 0 0 0... ...0 0 0 0 0 1 1 1 0 0 0 0 0... ...0 0 0 0 1 1 0 1 1 0 0 0 0... ...0 0 0 0 0 1 1 1 0 0 0 0 0... ...0 0 0 0 0 0 1 0 0 0 0 0 0... ...0 0 0 0 0 1 1 1 0 0 0 0 0... ...0 0 0 0 1 1 0 1 1 0 0 0 0... ...0 0 0 1 1 0 0 0 1 1 0 0 0... ...0 0 0 0 1 1 0 1 1 0 0 0 0... ...0 0 0 0 0 1 1 1 0 0 0 0 0... ...0 0 0 0 0 0 1 0 0 0 0 0 0... ...0 0 0 0 0 1 1 1 0 0 0 0 0... ...0 0 0 0 1 1 0 1 1 0 0 0 0... ...0 0 0 1 1 0 0 0 1 1 0 0 0... ...0 0 1 1 0 0 0 0 0 1 1 0 0... ...0 0 0 1 1 0 0 0 1 1 0 0 0... ...0 0 0 0 1 1 0 1 1 0 0 0 0... ...0 0 0 0 0 1 1 1 0 0 0 0 0... ...0 0 0 0 0 0 1 0 0 0 0 0 0... No. 2 Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 ...0 0 0 0 0 0 1 0 0 0 0 0 0... ...0 0 0 0 0 1 1 1 0 0 0 0 0... ...0 0 0 0 1 1 0 1 1 0 0 0 0... ...0 0 0 1 1 0 0 0 1 1 0 0 0... ...0 0 1 1 0 0 0 0 0 1 1 0 0... ...0 1 1 0 0 0 0 0 0 0 1 1 0... ...0 0 1 1 0 0 0 0 0 1 1 0 0... · · · · · · 3 This sequence has the form 1, 111, 11011, 111, 1, 1, 111, 11011, 1100011, 11011, 111, 1, | {z } | {z } 3 7 1, 111, 11011, 1100011, 110000011, 1100011, 11011, 111, 1, . . . | {z } 9 All digits from the above table generate an infinite matrix A. We described the elements of A. Let us take a Cartesian coordinate system C with origin in the point containing element ”1” in the topmost (i.e., the first) row of A. We assume that this row belongs to the ordinate axis of C (see Fig. 1) and that the points to the right of the origin have positive ordinates. The above digits generate an infinite sequence of squares, located in the half-plane (determined by C) where the abscissa of the points are nonnegative. Their diameters have the form ”110 . . . 011”. Exactly one of the diameters of each of considered square lies on the abscissa of C. It can be seen (and proved, e.g.,by induction) that the s-th square, denoted by Gs (s = 0, 1, 2, . . .) has a diameter with length 2s + 4 and the same square has a highest vertex with coordinates hs2 + 3s, 0i in C and a lowest vertex with coordinates hs2 + 5s + 4, 0i in C. Let us denote by ak,i an element of A with coordinates hk, ii in C. First, we determine the minimal nonnegative s for which the inequality s2 + 5s + 4 ≥ k holds. We denote it by s(k). Directly it is seen the following Lemma 2.1 The number s(k) admits the explicit representation:    0, if 0 ≤ k ≤ 4   h√ i   4k+9−5  , if k ≥ 5 and 4k + 9 is   2  s(k) = a square of an integer  i h√   4k+9−5  + 1, if k ≥ 5 and 4k + 9 is   2     not a square of an integer (2.1) 4 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 (s(k))2 + 3s(k) ≤ k ≤ (s(k))2 + 5s(k) + 4 (2.2) and the inequality hold. Second, we introduce the integer δ(k) and ε(k) by δ(k) ≡ k − (s(k))2 − 3s(k), (2.3) ε(k) ≡ (s(k))2 + 5s(k) + 4 − k. (2.4) From (2.2) we have δ(k) ≥ 0 and ε(k) ≥ 0. Let Pk be the infinite strip orthogonal to the abscissa of C and lying between the straight lines passing through those vertices of the square Gs(k) lying on the abscissa of C. Then δ(k) and ε(k) characterize the location of point with coordinate hk, ii in C in strip Pk . Namely, the following assertion is true. Proposition 2.1. The elements ak,i of the infinite matrix A are described as follows: if k ≤ (s(k))2 + 4s(k) + 2, then   0, if δ(k) < |i| or δ(k) ≥ |i| + 2, ak,i =  1, if |i| ≤ δ(k) ≤ |i| + 1 if k ≥ (s(k))2 + 4s(k) + 2, then ak,i   0, if ε(k) < |i| or ε(k) ≥ |i| + 2, =  1, if |i| ≤ ε(k) ≤ |i| + 1 where here and below s(k) is given by (2.1), δ(k) and ε(k) are given by (2.3) and (2.4), respectively. Below, we propose another description of elements of A, which can be proved (e.g., by induction) using the same considerations. ak,i   1, if hk, ii ∈       {h(s(k))2 + 3s(k), 0i, h(s(k))2 + 5s(k) + 4, 0i}    S   {h(s(k))2 + 3s(k) + j, −ji,       h(s(k))2 + 3s(k) + j, −j + 1i,       h(s(k))2 + 3s(k) + j, j − 1i,      h(s(k))2 + 3s(k) + j, ji : 1 ≤ j ≤ s(k) + 2} =   h(s(k))2 + 5s(k) + 4 − j, −ji,      h(s(k))2 + 5s(k) + 4 − j, −j + 1i,       h(s(k))2 + 5s(k) + 4 − j, j − 1i,       h(s(k))2 + 5s(k) + 4 − j, ji :       1 ≤ j ≤ s(k) + 1}     0, otherwise Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 5 Similar representations are possible for the ninth, tenth and eleventh problems. In [28] we introduce eight modifications of these problems, giving formulae for their (k, i)-th members ak,i . Essentially more interesting is Problem 103 from [10]: Smarandache numerical carpet: has the general form · · · 1 1 a 1 1 a 1 a 1 a 1 a b a 1 b c b a 1 b c d c b a b c d e d e f 1 a b c d 1 a b c d e f 1 a b c d 1 a b c d g e f c b e d e d e d 1 a b c d c b a 1 e d f 1 c c b a 1 b a 1 c b a 1 c b a 1 1 a b c b a 1 1 a a 1 b a 1 1 a 1 1 · · · On the border of level 0, the elements are equal to ”1”; they form a rhomb. Next, on the border of level 1, the elements are equal to ”a”; where ”a” is the sum of all elements of the previous border; the ”a”s form a rhomb too inside the previous one. Next again, on the border of level 2, the elements are equal to ”b”; 6 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 where ”b” is the sum of all elements of the previous border; the ”b”s form a rhomb too inside the previous one. And so on . . . The above square, that Smarandache named ”rhomb”, corresponds to the square from our construction for the case of s = 6, if we begin to count from s = 1, instead of s = 0. In [10] a particular solution of the Problem 103 is given, but there a complete solution is not introduced. We will give a solution below firstly for the case of Problem 103 and then for a more general case. It can be easily seen that the number of the elements of the s-th square side is s + 2 and therefore the number of the elements from the contour of this square is just equal to 4s + 4. The s-th square can be represented as a set of sub-squares, each one included in the next. Let us number them inwards, so that the outmost (boundary) square is the first one. As it is written in Problem 103, all of its elements are equal to 1. Hence, the value of the elements of the subsequent (second) square will be (using also the notation from problem 103): a1 = a = (s + 2) + (s + 1) + (s + 1) + s = 4(s + 1); the value of the elements of the third square will be a2 = b = a(4(s − 1) + 4 + 1) = 4(s + 1)(4s + 1); the value of the elements of the fourth square will be a3 = c = b(4(s − 2) + 4 + 1) = 4(s + 1)(4s + 1)(4s − 3); the value of the elements of the fifth square will be a4 = d = c(4(s − 3) + 4 + 1) = 4(s + 1)(4s + 1)(4s − 3)(4s − 7); etc.,where the square, corresponding to the initial square (rhomb), from Problem 103 has the form 1 · · · · · · a1 1 a2 · · · a2 a1 1 a2 a3 · · · a3 a2 a1 1 a1 a2 · · · a2 a1 1 1 a1 · · · a1 1 · · · 1 a1 1 a1 1 a1 1 1 Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 7 It can be proved by induction that the elements of this square that stay on t-th place are given by the formula t−2 Y at = 4(s + 1) (4s + 1 − 4i). i=0 If we would like to generalize the above problem, we can construct, e.g., the following extension: x x a1 x · · · · · · a1 x x a1 a2 · · · a2 a1 x a1 a2 a3 · · · a3 a2 a1 x a1 a2 · · · a2 a1 x x a1 · · · a1 x · · · x x where x is given number. Then we obtain a1 = 4(s + 1)x a2 = 4(s + 1)(4s + 1)x a3 = 4(s + 1)(4s + 1)(4s − 3)x a4 = 4(s + 1)(4s + 1)(4s − 3)(4s − 7)x etc. and for t ≥ 1 at = 4(s + 1) t−2 Y (4s + 1 − 4i)x. i=0 where it assumed that −1 Y · = 1. i=0 3. The results in this section are taken from [21]. The 15-th Smarandache’s problem from [10] is the following: “Smarandache’s simple numbers”: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, . . . A number n is called “Smarandache’s simple number” if the product of its proper divisors is less than or equal to n. Generally speaking, n has the form n = p, or n = p2 , or n = p3 , or n = pq, where p and q are distinct primes”. 8 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 Let us denote: by S - the sequence of all Smarandache’s simple numbers and by sn - the n-th term of S; by P - the sequence of all primes and by pn - the n-th term of P; by P 2 - the 3 3 ∞ sequence {p2n }∞ n=1 ; by P - the sequence {pn }n=1 ; by PQ - the sequence {p.q}p,q ∈ P, where p < q. In the present section we find πS (n) in an explicit form and using this, we find the n-th term of S in explicit form, too. First, we note that instead of πP (n) we use the notation π(n). Hence √ √ πP 2 (n) = π( n), πP 3 (n) = π( 3 n), Thus, using the definition of S, we get √ √ πS (n) = π(n) + π( n) + π( 3 n) + πPQ (n) (4.1) Our first aim is to express πS (n) in an explicit form. For π(n) some explicit formulae are proposed in [18]. Other explicit formulae for π(n) are given in [14]. One of them is known as Minác̈’s formula. It is given below π(n) = n X (k − 1)! + 1 (k − 1)! [ −[ ]]. k k (4.2) k=2 √ √ Therefore, the problem of finding of explicit formulae for functions π(n), π( n), π( 3 n) is solved successfully. It remains only to express πPQ (n) in an explicit form. √ Let k ∈ {1, 2, . . . , π( n)} be fixed. We consider all numbers of the kind pk q, which p ∈ P, q > pk for which pk .q ≤ n. The quality of these numbers is π( pnk ) − π(pk ), or which is the same π( n ) − k. pk (4.3) √ When k = 1, 2, . . . , π( n), the number pk .q, as defined above, describe all numbers of the kind p.q, with p, q ∈ P, p < q, p.q < n. But the quantity of the last numbers is equal to πPQ (n). Hence √ π( n) X n (π( ) − k), πPQ (n) = (4.4) pk k=1 because of (4.3). The equality (4.4), after a simple computation yields the formula √ π( n) πPQ (n) = X k=1 √ √ n π( n)(π( n) + 1) π( ) − . pk 2 (4.5) In [20] the identity π(b) X k=1 n n π( ) = π( ).π(b) + pk b n π( n 2 )−π( b ) X k=1 π( n ) pπ( nb )+k (4.6) is proved, under the condition b > 2 (b is a real number). When π( n2 ) = π( nb ), the right hand√ side of (4.6) is reduced to π( nb ).π(b). In the case b = n and n ≥ 4 equality (4.6) yields √ π( n) X k=1 √ n π( ) = (π( n))2 + pk √ π( n 2 )−π( n) X k=1 π( n ). pπ(√n)+k (4.7) Vol. 1 9 Remarks on Some of the Smarandache’s Problem. Part 2 If we compare (4.5) with (4.7) we obtain for n ≥ 4 π( n √ √ 2 )−π( X π( n)(π( n) − 1) + πPQ (n) = 2 √ n) k=1 π( n ). pπ(√n)+k (4.8) Thus, we have two different explicit representations for πPQ (n). These are formulae (4.5) √ √ π( n)(π( n)−1) and (4.8). We note that the right hand side of (4.8) reduces to , when π( n2 ) = 2 √ π( n). Finally, we observe that (4.1) gives an explicit representation for πS (n), since we may use formula (4.2) for π(n) (or other explicit formulae for π(n)) and (4.5), or (4.8) for πPQ (n). The following assertion solves the problem for finding of the explicit representation of sn . Theorem 4.1. The n-th term sn of S admits the following three different explicit representations: θ(n) sn = X [ k=0 1 1 + [ πSn(n) ] θ(n) sn = −2 X ζ(−2[ k=0 θ(n) sn = ]; (4.9) πS (n) ]); n X 1 k=0 Γ(1 − [ πSn(n) ]) , (4.10) (4.11) where n2 + 3n + 4 ], n = 1, 2, . . . 4 We note that (4.9)-(4.11) are representations using, respectively, “floor function”, Riemann’s Zeta-function and Euler’s Gamma-function. Also, we note that in (4.9)-(4.11) πS (k) is given by (4.1), π(k) is given by (4.2) (or by others formulae like (4.2)) and πPQ (n) is given by (4.5), or by (4.8). Therefore, formulae (4.9)-(4.11) are explicit. θ(n) ≡ [ 4. The results in this section are taken from [6]. The 17-th problem from [10] (see also the 22-nd problem from [16]) is the following: Smarandache’s digital products: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, | {z } | {z } 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, {z } | {z } | 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25 . . . | {z } | {z } (dp (n)is the product of digits.) Let the fixed natural number n have the form n = a1 a2 . . . ak , where a1 , a2 , . . . , ak ∈ {0, 1, . . . , 9} and a1 ≥ 1. Therefore, n= k X i=1 ai 10i−1 . 10 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 Hence, k = [log10 n] + 1 and a1 (n) ≡ a1 = [ a2 (n) ≡ a2 = [ a3 (n) ≡ a3 = [ n ], 10k−1 n − a1 10k−1 ], 10k−2 n − a1 10k−1 − a2 10k−2 ], 10k−3 ... a[log10 (n)] (n) ≡ ak−1 = [ n − a1 10k−1 − . . . − ak−2 102 ], 10 a[log10 (n)]+1 (n) ≡ ak = n − a1 10k−1 − . . . − ak−1 10. Obviously, k, a1 , a2 , . . . , ak are functions only of n. Therefore, [log10 (n)]+1 dp (n) = Y ai (n). i=1 5. The results in this section are taken from [4, 27]. The 20-th problem from [10] is the following (see also Problem 25 from [16]): Smarandache devisor products: 1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, 225, 1024, 17, 5832, 19, 8000, 441, 484, 23, 331776, 125, 676, 729, 21952, 29, 810000, 31, 32768, 1089, 1156, 1225, 10077696, 37, 1444, 1521, 2560000, 41, . . . (Pd (n) is the product of all positive divisors of n.) The 21-st problem from [10] is the following (see also Problem 26 from [16]): Smarandache proper devisor products: 1, 1, 1, 2, 1, 6, 1, 8, 3, 10, 1, 144, 1, 14, 15, 64, 1, 324, 1, 400, 21, 22, 1, 13824, 5, 26, 27, 784, 1, 27000, 1, 1024, 33, 34, 35, 279936, 1, 38, 39, 64000, 1, . . . (pd (n) is the product of all positive divisors of n but n.) Let us denote by τ (n) the number of all devisors of n. It is well-known (see, e.g., [13]) that p Pd (n) = nτ (n) (6.1) and of course, we have Pd (n) . (6.2) n But (6.1) is not a good formula for Pd (n), because it depends on function τ and to express τ (n) we need the prime number factorization of n. pd (n) = Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 11 Below, we give other representations of Pd (n) and pd (n), which do not use the prime number factorization of n. Proposition 6.1. For n ≥ 1 representation Pd (n) = n Y n k [ k ]−[ n−1 k ] (6.3) k=1 holds. Here and further the symbols Y X · and k/n · k/n mean the product and the sum, respectively, of all divisors of n. Let n−1 n ] θ(n, k) ≡ [ ] − [ k k   1, if k is a divisor of n =  0, otherwise The following assertion is obtained as a corollary of (6.2) and (6.3). Proposition 6.2. For n ≥ 1 representation pd (n) = n−1 Y n k [ k ]−[ n−1 k ] k=1 holds. For n = 1 we have pd (1) = 1. Proposition 6.3. For n ≥ 1 representation Pd (n) = n Y [ nk ]! [ n−1 k ]! k=1 holds, where here and further we assume that 0! = 1. Now (6.2) and (6.5) yield. Proposition 6.4. For n ≥ 2 representation n Y [ nk ]! pd (n) = n−1 [ k ]! k=2 holds. Another type of representation of pd (n) is the following Proposition 6.5. For n ≥ 3 representation pd (n) = n−2 Y k=1 (k!)θ(n,k)−θ(n,k+1) , (6.5) 12 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 where θ(n, k) is given by (6.4). Further, we need the following Theorem 6.1. [22] For n ≥ 2 the identity n−1 n Y Y n n n (k!)[ k ]−[ k+1 ] [ ]! = k (6.6) k=1 k=2 holds. Now, we shall deduce some formulae for n Y Pd (k) n Y and k=1 pd (k). k=1 Proposition 6.6. Let f be an arbitrary arithmetic function. then the identity n Y (Pd (k))f (k) = k=1 n Y k ρ(n,k) (6.7) k=1 holds, where n ρ(n, k) = [k] X f (ks). s=1 Now we need the following Lemma 6.1. For n ≥ 1 the identity n n Y Y n n k[ k ] [ ]! = k k=1 k=1 holds. Proposition 6.7. For n ≥ 1 the identity n Y Pd (k) = n Y n [ ]! k (6.8) k=1 k=1 holds. As a corollary from (6.2) and (6.8), we also obtain Proposition 6.8. For n ≥ 2 the identity n Y pd (k) = k=1 n Y n [ ]! k (6.9) k=2 holds. From (6.6) and (6.9), we obtain Proposition 6.9. For n ≥ 2 the identity n Y k=1 pd (k) = n−1 Y n n (k!)[ k ]−[ k+1 ] k=1 holds. As a corollary from (6.10) we obtain, because of (6.2) (6.10) Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 13 Proposition 6.10. For n ≥ 1 the identity n Y n Y Pd (k) = n n (k!)[ k ]−[ k+1 ] (6.10) k=1 k=1 holds. Now, we return to (6.7) and suppose that f (k) > 0 (k = 1, 2, . . .). Then after some simple computations we obtain Proposition 6.11. For n ≥ 1 representation Pd (k) = n Y k σ(n,k) (6.11) k=1 holds, where P[ nk ] s=1 σ(n, k) = P[ n−1 k ] f (ks) − s=1 f (ks) . f (n) For n ≥ 2 representation pd (k) = n−1 Y k σ(n,k) (6.12) k=1 holds. Note that although f is an arbitrary arithmetic function, the situation with (6.11) and (6.12) is like the case f (x) ≡ 1, because  P[ nk ] P[ n−1  1, if k is a divisor of n k ] s=1 f (ks) − s=1 f (ks) =  0, otherwise f (n) Finally, we use (6.7) to obtain some new inequalities, involving Pd (k) and pd (k) for k = 1, 2, . . . , n. Putting n X F (n) = f (k) k=1 we rewrite (6.7) as n Y (Pd (k)) f (k) F (n) = n Y k( P[ n ] k s=1 f (ks))/(F (n)) . k=1 k=1 Then we use the well-known Jensen’s inequality n X αk xk ≥ k=1 n Y k xα k , k=1 that is valid for arbitrary positive numbers xk , αk (k = 1, 2, . . . , n) such that n X k=1 αk = 1, (6.13) 14 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 for the case: xk = Pd (k), αk = f (k) . F (n) Thus we obtain from (6.13) inequality n X f (k).Pd (k) ≥ ( k=1 n X f (k)). k=1 n Y k( P[ n ] k s=1 f (ks))/( Pn s=1 f (s)) . (6.14) k=1 If f (x) ≡ 1, then (6.14) yields the inequality n n √ Y n 1X n Pd (k) ≥ ( k)[ k ] . n k=1 k=1 If we put in (6.14) f (k) = g(k) k for k = 1, 2, . . . , n, then we obtain n X g(k).pd (k) ≥ ( k=1 n X g(k) k=1 k ). n √ Y P[ n ] k k ( k)( s=1 P g(ks) g(s) )/( n s=1 s s ) . (6.15) k=1 because of (6.2). Let g(x) ≡ 1. Then (6.15) yields the very interesting inequality ( n n √ Y 1 X k H n pd (k))Hn ≥ ( k) [ k ] , Hn k=1 k=1 where Hm denotes the m-th partial sum of the harmonic series, i.e., Hm = 1 1 1 + + ... + . 1 2 m All of the above inequalities become equalities if and only if n = 1. 6. The results in this section are taken from [29]. The 25-th and the 26-th problems from [10] (see also the 30-th and the 31-st problems from [16]) are the following: Smarandache’s cube free sieve: 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, . . . Definition: from the set of natural numbers (except 0 and 1): - take off all multiples of 23 (i.e. 8,16,24,32,40,. . . ) Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 15 - take off all multiples of 33 - take off all multiples of 53 . . . and so on (take off all multiples of all cubic primes). Smarandache’s m-power free sieve: Definition: from the set of natural numbers (except 0 and 1) take off all multiples of 2m , afterwards all multiples of 3m . . . and so on(take off all multiples of all m-power primes, m ≥ 2). (One obtains all m-power free numbers.) Here we introduce the solution for both of these problems. (m) (m) (m) For every natural number m we denote the increasing sequence a1 , a2 , a3 , . . . of all m-power free numbers by m. Then we have ∅ ≡ 1 ⊂ 2 ⊂ . . . ⊂ (m − 1) ⊂ m ⊂ (m + 1) ⊂ . . . Also, for m ≥ 2 we have m= m−1 [ (2)k k=1 where (2)k = {x|(∃x1 , . . . , xk ∈ 2)(x = x1 .x2 . . . xk )} for each natural number k ≥ 1. Let us consider m as an infinite sequence for m = 2, 3, . . .. Then 2 is a subsequence of m. Therefore, the inequality a(m) ≤ a(2) n n holds for n = 1, 2, 3, . . . . Let p1 = 2, p2 = 3, p3 = 5, p4 = 7, . . . be the sequence of all primes. It is obvious that this sequence is a subsequence of 2. Hence the inequality a(2) n ≤ pn holds for n = 1, 2, 3, . . .. But it is well-known that pn ≤ θ(n) ≡ [ n2 + 3n + 4 ] 4 (see [12]). Therefore, for any m ≥ 2 and n = 1, 2, 3, . . . we have a(m) ≤ a(2) n n ≤ θ(n). Hence, there exits λ(n) such that λ(n) ≤ θ(n) and inequality: a(m) ≤ a(2) n n ≤ λ(n). holds. In particular, it is possible to use θ(n) instead of λ(n). (m) In [28] we find the following explicit formulae for an when m ≥ 2 is fixed: λ(n) a(m) = n X k=0 [ 1 1 + [ πmn(k) ] ]; (7.1) 16 Mladen V. Vassilev, Missana and Krassimir T. Atanassov λ(n) a(m) = −2 n X ζ(−2[ k=0 λ(n) a(m) n = πm (k) ]); n X 1 k=0 Γ(1 − [ πmn(k) ]) . No. 2 (7.2) (7.3) Thus, the 26-th Smarandache’s problem is solved and for m = 3 the 25-th Smarandache’s problem is solved, too. The following problems are interesting. Problem 7.1. Does there exist a constant C > 1, such that λ(n) ≤ C.n? Problem 7.2. Is C ≤ 2? Below we give the main explicit representation of function πm (n), that takes part in for(m) mulae (7.1) - (7.3). In this way we find the main explicit representation for an , that is based on formulae (7.1) - (7.3), too. Theorem 7.1. Function πm (n) allows representation X n πm (n) = n − 1 + (−1)ω(s) .[ m ], s T √ m s∈2 {2,3,...,[ n]} where ω(s) denotes the number of all different prime divisors of s. 7. The results in this section are taken from [24]. The 28-th problem from [10] (see also the 94-th problem from [16]) is the following: Smarandache odd sieve: 7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 83, 85, 91, 93, 97, . . . (All odd numbers that are not equal to the difference of two primes). A sieve is to get this sequence: - subtract 2 from all prime numbers and obtain a temporary sequence; - choose all odd numbers that do not belong to the temporary one. We find an explicit form of the n-th term of the above sequence, that will be denoted by C = {Cn }∞ n=1 below. Firstly, we shall note that the above definition of C can be interpreted to the following equivalent form as follows, having in mind that every odd number is a difference of two prime numbers if and only if it is a difference of a prime number and 2: Smarandache’s odd sieve contains exactly these odd numbers that cannot be represented as a difference of a prime and 2. We rewrite the last definition to the following equivalent form, too: Smarandache’s odd sieve contains exactly these odd numbers that are represented as a difference of a composite odd number and 2. We find an explicit form of the n-th term of the above sequence, using the third definition of it. Initially, we use the following two assertions. Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 17 Lemma 8.1. For every natural number n ≥ 1, Cn+1 is exactly one of the numbers: u ≡ Cn + 2, v ≡ Cn + 4 or w ≡ Cn + 6. Corollary 8.1. For every natural number n ≥ 1: Cn+1 ≤ Cn + 6. Corollary 8.2. For every natural number n ≥ 1: Cn ≤ 6n + 1. (8.1) Now, we return to the Smarandache’s problem. In [18] the following three universal explicit formulae are introduced, using numbers πC (k) (k = 0, 1, 2, . . .), that can be used to represent numbers Cn : ∞ X Cn = [ k=0 Cn = −2 1 1 + [ πCn(k) ] ∞ X ζ(−2[ k=0 ], πC (k) ]), n ∞ X 1 k=0 Γ(1 − [ πCn(k) ]) Cn = . For the present case, having in mind (8.1), we substitute symbol ∞ with 6n + 1 in sum k=0 for Cn and we obtain the following sums: P∞ Cn = 6n+1 X [ k=0 Cn = −2 6n+1 X 1 1 + [ πCn(k) ] ζ(−2[ k=0 Cn = ], (8.2) πC (k) ]), n 6n+1 X 1 k=0 Γ(1 − [ πCn(k) ]) . (8.3) (8.4) We must explain why πC (n) (n = 1, 2, 3, . . .) is represented in an explicit form. It can be directly seen that the number of the odd numbers, that are not bigger than n, is exactly equal to n α(n) = n − [ ], (8.5) 2 because the number of the even numbers that are not greater than n is exactly equal to [ n2 ]. Let us denote by β(n) the number of all odd numbers not bigger than n, that can be represented as a difference of two primes. According to the second form of the above given definition, β(n) coincides with the number of all odd numbers m such that m ≤ n and m has the form m = p − 2, where p is an odd prime number. Therefore, we must study all odd prime numbers, because of the inequality m ≤ n. The number of these prime numbers is exactly π(n + 2) − 1. therefore, β(n) = π(n + 2) − 1. (8.6) 18 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 Omitting from the number of all odd numbers that are not greater than n the quantity of those numbers that are a difference of two primes, we find exactly the quantity of these odd numbers that are not greater than n and that are not a difference of two prime numbers, i.e., πC (n). Therefore, the equality πC (n) = α(n) − β(n) holds and from (8.5) and (8.6) we obtain: n n πC (n) = (n − [ ]) − (π(n + 2) − 1) = n + 1 − [ ] − π(n + 2). 2 2 But π(n + 2) can be represented in an explicit form, e.g., by Minác̈’s formula and therefore, we obtain that the explicit form of πC (N ) is πC (N ) = N + 1 − [ N +2 X N (k − 1)! + 1 (k − 1)! ]− [ −[ ]], 2 k k (8.7) k=2 where N ≥ 1 is a fixed natural number. It is possible to put [ N2+3 ] instead of N + 1 − [ N2 ] into (8.7). Now, using each of the formulae (8.2) - (8.4), we obtain Cn in an explicit form, using (8.7). It can be checked directly that C1 = 7, C2 = 13, C3 = 19, C4 = 23, C5 = 25, C6 = 31, C7 = 33, . . . and πC (0) = πC (1) = πC (2) = πC (3) = πC (4) = πC (5) = πC (6) = 0. Therefore from (8.2) - (8.4) we have the following explicit formulae for Cn Cn = 7 + 6n+1 X [ k=7 Cn = 7 + −2 6n+1 X 1 1 + [ πCn(k) ] ζ(−2.[ k=7 Cn = 7 + ], πC (k) ]), n 6n+1 X 1 k=7 Γ(1 − [ πCn(k) ]) , where πC (k) is given by (8.7). 8. The results in this section are taken from [7, 26]. The 46-th Smarandache’s problem from [10] is the following: Smarandache’s prime additive complements; 1, 0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 5, 4, 3, 2, 1, 0, 1, 0, 5, 4, 3, 2, 1, 0, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0 . . . Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 19 (For each n to find the smallest k such that n + k is prime.) Remarks: Smarandache asked if it is possible to get as large as we want but finite decreasing k, k − 1, k − 2, . . . , 2, 1, 0 (odd k) sequence included in the previous sequence - i.e., for any even integer are there two primes those difference is equal to it? He conjectured the answer is negative. Obviously, the members of the above sequence are differences between first prime number that is greater or equal to the current natural number n and the same n. It is well-known that the number of primes smaller than or equal to n is π(n). Therefore, the prime number smaller than or equal to n is pπ(n) . Hence, the prime number that is greater than or equal to n is the next prime number, i.e., pπ(n)+1 . Finally, the n-th member of the above sequence will be equal to   p π(n)+1 − n, if n is not a prime number  0, otherwise We shall note that in [3] the following new formula pn for every natural number n is given: θ(n) pn = X sg(n − π(i)), i=0 where θ(n) = [ n 2 +3n+4 ] 4 and   0, if x ≤ 0, sg(x) =  1, if x > 0. Let us denote by an the n-th term of the above sequence. Next, we propose a way for obtaining an explicit formula for an (n = 1, 2, 3, . . .). Extending the below results, we give an answer to the Smarandache’s question from his own remark in [10]. At the end, we propose a generalization of Problem 46 and present a proof of an assertion related to Smarandache’s conjecture for Problem 46. Proposition 9.1. an admits the representation an = pπ(n−1)+1 − n, (9.1) where n = 1, 2, 3, . . ., π is the prime counting function and pk is the k-th term of prime number sequence. It is clear that (9.1) gives an explicit representation for an since several explicit formulae for π(k) and pk are known (see, e.g. [14]). Let us define n(m) = m! + 2. Then all numbers n(m), n(m) + 1, n(m) + 2, . . . , n(m) + m − 2 are composite. Hence an(m) ≥ m − 1. 20 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 This proves the Smarandache’s conjecture, since m may grow up to infinity. Therefore {an }∞ n=1 is unbounded sequence. Now, we shall generalize Problem 46. Let c ≡ c1 , c2 , c3 , . . . be a strictly increasing sequence of positive integers. Definition. Sequence b ≡ b1 , b 2 , b 3 , . . . is called c-additive complement of c if and only if bn is the smallest non-negative integer, such that n + bn is a term of c. The following assertion generalizes Proposition 1. Proposition 9.2. bn admits the representation bn = cπc (n−1)+1 − n, where n = 1, 2, 3, . . ., πc (n) is the counting function of c, i.e., πc (n) equals to the quantity of cm , m = 1, 2, 3, . . ., such that cm ≤ n. Let dn ≡ cn+1 − cn (n = 1, 2, 3, . . .). The following assertion is related to the Smarandache’s conjecture from Problem 46. ∞ Proposition 9.3. If {dn }∞ n=1 is unbounded sequence, then {bn }n=1 is unbounded sequence, too. Open Problem. Formulate necessary conditions for the sequence {bn }∞ n=1 to be unbounded. 9. The results in this section are taken from [23]. Solving of the Diophantine equation 2x2 − 3y 2 = 5 (10.1) i.e., 2x2 − 3y 2 − 5 = 0 was put as an open Problem 78 by F. Smarandache in [16]. In [28] this problem is solved completely. Also, we consider here the Diophantine equation l2 − 6m2 = −5, i.e., l2 − 6m2 + 5 = 0 and the Pellian equation u2 − 6v 2 = 1, i.e., u2 − 6v 2 − 1 = 0. Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 21 In [28] we introduce a generalization of the Smarandache’s problem 78 from [16]. If we consider the Diophantine equation 2x2 − 3y 2 = p, (10.2) where p 6= 2 is a prime number, then using [13], Chapter VII, exercise 2 and the same method as in the case of (10.1), we obtain the following result. Theorem 10.1. (1) The necessary and sufficient condition for solvability of (10.2) is: p ≡ 5(mod24) or p ≡ 23(mod24) (10.3) (2) if (10.3) is valid, then there exist exactly one solution < x, y >∈ N 2 of (10.2) such that the inequalities r 3 x< .p 2 and r 3 y< .p 2 holds. Every other solution < x, y >∈ N 2 of (10.2) has the form: x = l + 3m y = l + 2m, where < l, m >∈ N 2 is a solution of the Diophantine equation l2 − 6m2 = −p. The problem how to solve the Diophantine equation, a special case of which is the above one, is considered in Theorem 110 from [13]. 10. The results in this section are taken from [9]. In [15, 17] F. Smarandache formulates the following four problems: Problem 1. Let p be an integer ≥ 3. Then: p is a prime if and only if (p − 3)! is congruent to p−1 (modp). 2 Problem 2. Let p be an integer ≥ 4. Then: p is a prime if and only if p (p − 4)! is congruent to (−1)d 3 e+1 d p+1 e(modp). 6 Problem 3. Let p be an integer ≥ 5. Then: p is a prime if and only if (11.1) 22 Mladen V. Vassilev, Missana and Krassimir T. Atanassov (p − 5)! is congruent to rh + r2 − 1 (modp), 24 No. 2 (11.2) p with h = d 24 e and r = p − 24. Problem 4. Let p = (k − 1)!h + 1 be a positive integer k > 5, h natural number. Then: p is a prime if and only if (p − k)! is congruent to (−1)t h(modp). (11.3) with t = h + d hp e + 1. Everywhere above dxe means the inferior integer part of x, i.e., the smallest integer greater than or equal to x. In [28] we discussed these four problems. Problem 1. Admits the following representation: Let p ≥ 3 be an odd number. Then: p is a prime if and only if (p − 3)! ≡ p−1 (modp). 2 Different than Smarandache’s proof of this assertion is given in [28]. Problem 2. Is false, because, for example, if p = 7, then (11.1) obtains the form 6 ≡ (−1)4 2(mod7), where 6 = (7 − 4)! and 7 8 (−1)4 2 = (−1)d 3 e+1 d e, 6 i.e., 6 ≡ 2(mod7), which is impossible. Problem 3. Can be modified, having in mind that from r = p − 24h it follows: r2 − 1 p2 − 48ph + 242 h2 − 1 = (p − 24h).h + 24 24 p2 − 1 p2 − 1 2 2 = ph − 24h + − 2ph + 24h = − ph, 24 24 rh + i.e., (11.2) has the form p is a prime if and only if (p − 5)! is congruent to p2 − 1 (modp), 24 Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 23 Different than the Smarandache’s proof of this assertion is given in [28]. Problem 4. Also can be simplified, because p t=h+d e+1 h (k − 1)!h + 1 e+1 h = h + (k − 1)! + 1 + 1 = h + (k − 1)! + 2, =h+d i.e., (−1)t = (−1)h , because for k > 2: (k − 1)! + 2 is an even number. Therefore, (11.3) obtains the form p is a prime if and only if (p − k)! is congruent to (−1)h h(modp), Let us assume that (11.4) is valid. We use again the congruences (p − 1) ≡ −1(modp) (p − 2) ≡ −2(modp) ... (p − (k − 1)) ≡ −(k − 1)(modp) and obtain the next form of (11.4) p is a prime if and only if (p − 1)! ≡ (−1)h .(−1)k−1 .(k − 1)!.h(modp) or p is a prime if and only if (p − 1)! ≡ (−1)h+k−1 .(p − 1)(modp). But the last congruence is not valid, because, e.g., for k = 5, h = 3, p = 73 = (5 − 1)! + 11 holds 72! ≡ (−1)9 .72(mod73), 2 i.e., 72! ≡ 1(mod73), 1 In 2 In [28] there is a misprint: 3! instead of 3. [28] there is a misprint: (−1)9 instead of (−1)7 . 24 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 while from Wilson’s Theorem follows that 72! ≡ −1(mod73). 11. The results in this section are taken from [5]. In [17] F. Smarandache discussed the following particular cases of the well-known characteristic functions (see, e.g., [11, 30]). 12.1) Prime function: P : N → {0, 1}, with   0, if n is a prime P (n) =  1, otherwise More generally: Pk : N k → {0, 1}, where k ≥ 2 is an integer, and   0, if n , n , . . . , n are all prime numbers 1 2 k Pk (n1 , n2 , . . . , nk ) =  1, otherwise 12.2) Coprime function is defined similarly: Ck : N k → {0, 1}, where k ≥ 2 is an integer, and   0, if n , n , . . . , n are coprime numbers 1 2 k Ck (n1 , n2 , . . . , nk ) =  1, otherwise In [28] we formulate and prove four assertions related to these functions. Proposition 12.1. For each k, n1 , n2 , . . . , nk natural numbers: Pk (n1 , n2 , . . . , nk ) = 1 − k Y (1 − P (ni )). i=1 Proposition 12.2. For each k, n1 , n2 , . . . , nk natural numbers: Ck (n1 , n2 , . . . , nk ) = 1 − k k Y Y (1 − C2 (ni , nj )). i=1 j=i+1 Proposition 12.3. For each natural number n: Cπ(n)+P (n) (p1 , p2 , . . . , pπ(n)+P (n)−1 , n) = P (n). Proposition 12.4. For each natural number n: π(n)+P (n)−1 P (n) = 1 − Y (1 − C2 (pi , n)). i=1 Corollary 12.1. For each natural number k, n1 , n2 , . . . , nk : Pk (n1 , n2 , . . . , nk ) = 1 − k π(ni )+P Y Y(ni )−1 i=1 j=1 (1 − C2 (pj , ni )). Vol. 1 Remarks on Some of the Smarandache’s Problem. Part 2 25 References [1] K.Atanassov, On some of the Smarandache’s problems, American Research Press, Lupton, 1999. [2] K.Atanassov, Remarks on some of the Smarandache’s problems. Part 1, Smarandache Notions Journal, Spring, 12(2001), 82-98. [3] K.Atanassov, A new formula for the n-th prime number. Comptes Rendus de l’Academie Bulgare des Sciences,7-8-9(2001). [4] K.Atanassov, On the 20-th and the 21-st Smarandache’s Problems. Smarandache Notions Journal, Spring ,1-2-3(2001), 111-113. [5] K.Atanassov, On four prime and coprime functions, Smarandache Notions Journal, Spring ,1-2-3(2001), 122-125. [6] K.Atanassov, On the 17-th Smarandache’s Problem, Smarandache Notions Journal, Spring ,1-2-3(2002), 124-125. [7] K.Atanassov, On the 46-th Smarandache’s Problem, Smarandache Notions Journal, Spring ,1-2-3(2002), 126-127. [8] K.Atanassov, On the second Smarandache’s Problem, Notes on Number Theory and Discrete Mathematics, 3(2003), 46-48. [9] K.Atanassov, On four Smarandache’s Problems, Notes on Number Theory and Discrete Mathematics,1-6(2005). [10] Dumitrescu C., V.Seleacu, Some Sotions and Questions in Number Theory, Erhus Univ. Press, Glendale, 1994. [11] Grauert H., Lieb I., Fischer W, Differential- und Integralrechnung, Springer-Verlag, Berlin, 1967. [12] Mitrinović, D., M. Popadić, Inequalities in Number Theory, Niś, Univ. of Niś, 1978. [13] Nagell T., Introduction to Number Theory, John Wiley ＆ Sons, Inc., New York, 1950. [14] Ribenboim P. The New Book of Prime Number Records, Springer, New York, 1995. [15] F.Smarandache, Criteria for a number to be prime, Gazeta Matematica, Vol. XXXVI, No. 2, 49-52, Bucharest, 1981 (in Romanian). [16] F.Smarandache, Only Problems, Not Solutions!, Xiquan Publ. House, Chicago, 1993. [17] F.Smarandache, Collected Papers, Vol. II, Kishinev University Press, Kishinev, 1997. [18] Vassilev-Missana M., Three formulae for n-th prime and six for n-th term for twin primes, Notes on Number Theory and Discrete Mathematics, 1(2001), 15-20. [19] Vassilev-Missana M., Some explicit formulae for the composite numbers. Notes on Number Theory and Discrete Mathematics, 2 (2001). [20] Vassilev-Missana M., On one remarkable identity related to function π(x), Notes on Number Theory and Discrete Mathematics, 4 (2001), 129-136. [21] Vassilev-Missana M., On 15-th Smarandache’s problem. Notes on Number Theory and Discrete Mathematics, 2 (2003) 42-45. [22] Vassilev-Missana M., Some representations concerning the product of divisors of n. Notes on Number Theory and Discrete Mathematics, 2 (2004), 54-56. [23] Vassilev M., Atanassov K., Note on the Diophantine equation 2x2 − 3y 2 = p, Smarandache Notions Journal, 1-2-3 (2000) 64-68. 26 Mladen V. Vassilev, Missana and Krassimir T. Atanassov No. 2 [24] Vassilev-Missana M., K.Atanassov, On 28-th Smarandache’s Problem. Notes on Number Theory and Discrete Mathematics, 2 (2001), 61-64. [25] Vassilev-Missana M., K.Atanassov, On five Smarandache’s problems. Notes on Number Theory and Discrete Mathematics, Vol. 10, 2 (2004), 34-53. [26] Vassilev-Missana, M., K. Atanassov, Remarks on the 46-th Smarandache’s Problem. Notes on Number Theory and Discrete Mathematics, 3 (2004), 84-88. [27] Vassilev-Missana, M., K. Atanassov, On two Smarandache’s problems, Notes on Number Theory and Discrete Mathematics, 4 (2004), 106-112. [28] Vassilev-Missana, M., K. Atanassov, Some Smarandache problems, Hexis, Phoenix, 2004. [29] Vassilev P., Vassilev-Missana M., K.Atanassov, On 25-th and 26-st Smarandache’s Problems. Notes on Number Theory and Discrete Mathematics, 4 (2003), 99-104. [30] Yosida K., Function Analysys, Springer-Verlag, Berlin, 1965. Scientia Magna Vol. 1 (2005), No. 2, 27-35 Smarandache groupoids W.B.Vasantha Kandasamy Department of Mathematics, Indian Institute of Technology, Madras Chennai-600 036, India vasantak@md3.vsnl.net.in Abstract In this paper, we study the concept of Smarandache groupoids, subgroupoids, ideal of groupoids, semi-normal subgroupoids, Smarandache-Bol groupoids and Strong Bol groupoids and obtain many interesting results about them. Keywords Smarandache groupoid; Smarandache subgroupoid; Smarandache ideal of a Smarandache groupoid; Smarandache semi-normal groupoid; Smarandache normal groupoid; Smarandache semi conjugate subgroupoid; Smarandache Bol groupoid; Smarandache Moufang groupoid. Definition [1]: A groupoid (G, ∗) is a non-empty set, closed with respect to an operation ∗ (in general ∗ need not to be associative). Definition 1: A Smarandache groupoid G is a groupoid which has a proper subset S ⊂ G which is a semigroup under the operation of G. Example 1: Let (G, ∗) be a groupoid on modulo 6 integers. G = {0, 1, 2, 3, 4, 5} is given by the following table: ∗ 0 1 2 3 4 5 0 0 3 0 3 0 3 1 1 4 1 4 1 4 2 2 5 2 5 2 5 3 3 0 3 0 3 0 4 4 1 4 1 4 1 5 5 2 5 2 5 2 Clearly S1 = {0, 3}, S2 = {1, 4} and S3 = {2, 5} are semigroups of G. So (G, ∗) is a Smarandache groupoid. Example 2: Let G = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} be the set of integers modulo 10. Define an operation ∗ on G by choosing a pair (1, 5) such that a ∗ b = 1a + 5b(mod 10) for all a, b ∈ G. The groupoid is given by the following table. 28 W.B.Vasantha Kandasamy No. 2 ∗ 0 1 2 3 4 5 6 7 8 9 0 0 5 0 5 0 5 0 5 0 5 1 1 6 1 6 1 6 1 6 1 6 2 2 7 2 7 2 7 2 7 2 7 3 3 8 3 8 3 8 3 8 3 8 4 4 9 4 9 4 9 4 9 4 9 5 5 0 5 0 5 0 5 0 5 0 6 6 1 6 1 6 1 6 1 6 1 7 7 2 7 2 7 2 7 2 7 2 8 8 3 8 3 8 3 8 3 8 3 9 9 4 9 4 9 4 9 4 9 4 Clearly S1 = {0, 5}, S2 = {1, 6}, S3 = {2, 7}, S4 = {3, 8} and S5 = {4, 9} are semigroupoids under the operation ∗. Thus {G, ∗, (1, 5)} is a Smarandache groupoid. Theorem 2. Let Z2p = {0, 1, 2, · · · , 2p−1}. Define ∗ on Z2p for a, b ∈ Z2p by a∗b = 1a+pb( mod 2p). {Z2p , ∗, (1, p)} is a Smarandache groupoid. Proof . Under the operation ∗ defined on Z2p we see S1 = {0, p}, S2 = {1, p + 1}, S3 = {2, p + 2},· · · ,Sp = {p − 1, 2p − 1} are semigroup under the operation ∗. Hence {Z2p , ∗, (1, p)} is a Smarandache groupoid. Example 3: Take Z6 = {0, 1, 2, 3, 4, 5}. (2, 5) = (m, n). For a, b ∈ Z6 define a ∗ b = ma + nb(mod 6). The groupoid is given by the following table: ∗ 0 1 2 3 4 5 0 0 5 4 3 2 1 1 2 1 0 5 4 3 2 4 3 2 1 0 5 3 0 5 4 3 2 1 4 2 1 0 5 4 3 5 4 3 2 1 0 5 Every singleton is an idempotent semigroup of Z6 . Theorem 3. Let Z2p = {0, 1, 2, · · · , p − 1}. Define ∗ on Z2p by a ∗ b = 2a + (2p − 1)b(mod 2p) for a, b ∈ Z2p . Then {Z2p , ∗, (2, 2p − 1)} is a Smarandache groupoid. Proof . Under the operation ∗ defined on Z2p we see that every element of Z2p is idempotent, therefore every element forms a singleton semigroup. Hence the claim. Example 4: Consider Z6 = {Z6 , ∗, (4, 5)} given by the following table: Vol. 1 29 Smarandache groupoids ∗ 0 1 2 3 4 5 0 0 5 4 3 2 1 1 4 3 2 1 0 5 2 2 1 0 5 4 3 3 0 5 4 3 2 1 4 4 3 2 1 0 5 5 2 1 0 5 4 3 {3} is a semigroup. Hence ∗ is a Smarandache groupoid. It is easily verified that Z6 is a Smarandache groupoid as {Z6 , ∗, (4, 5)} has an idempotent semigroup {3} under ∗. Theorem 4. Let Z2p = {0, 1, 2, · · · , 2p − 1} be the set of integers modulo 2p. Define ∗ on a, b ∈ Z2p by a(2p − 2) + b(2p − 1)(mod 2p). Then {Z2p , ∗, (2p − 2, 2p − 1)} is a Smarandache groupoid. Proof . Z2p = {0, 1, 2, · · · , 2p − 1}. Take (2p − 2, 2p − 1) = 1 from Z2p . For a, b ∈ Zp define a∗b = a(2p−2)+b(2p−1)(mod 2p). Clearly for a = b = p we have (2p−2)p+(2p−1)p = p(mod 2p). Hence {p} is an idempotent semigroup of Z2p . So {Z2p , ∗, (2p−2, 2p−1)} is a Smarandache groupoid. Definition 5: Let (G, ∗) be a Smarandache groupoid. A non-empty subset H of G is said to be a Smarandache groupoid if H contains a proper subset K ⊂ H such that K is a semigroup under the operation ∗. Theorem 6. Not every subgroupoid of a Smarandache groupoid S is in general a Smarandache subgroupoid of S. Proof . By an example. Let Z6 = {0, 1, 2, 3, 4, 5}(mod 6). Take (t, u) = (4, 5) = 1. For a, b ∈ Z6 define ∗ on Z6 by a ∗ b = at + bu(mod 6) given by the following table: ∗ 0 1 2 3 4 5 0 0 5 4 3 2 1 1 4 3 2 1 0 5 2 2 1 0 5 4 3 3 0 5 4 3 2 1 4 4 3 2 1 0 5 5 2 1 0 5 4 3 Clearly {Z6 , ∗, (4, 5)} is a Smarandache groupoid for it contains {0, 3} as a semigroup. But this groupoid has the following subgroupoids: A1 = {0, 2, 4} and A2 = {1, 3, 5}. A1 has no non-trivial semigroup({0} is a trivial semigroup). But A2 has a non-trivial semigroup, viz.{3}. Hence the claim. Theorem 7. If a groupoid contains a Smarandache subgroupoid, then the groupoid is a Smarandache groupoid. 30 W.B.Vasantha Kandasamy No. 2 Proof . Let G be a groupoid and H ⊂ G be a Smarandache subgroupoid, that is H contains a proper subset P ⊂ H such that P is a semigroup. So P ⊂ G and P is a semigroup. Hence G is a Smarandache groupoid. Definition 8: i) A Smarandache Left Ideal A of the Smarandache Groupoid G satisfies the following conditions: 1. A is a Smarandache subgroupoid. 2. For all x ∈ G, and x ∈ A, xa ∈ A. ii) Similarly, one defines a Smarandache Right Ideal. iii) If A is both a Smarandache right and left ideals then A is a Smarandache Ideal. We take {0} as a trivial Smarandache ideal. Example 5: Let {Z6 , ∗, (4, 5)} be a Smarandache groupoid. A = {1, 3, 5} is a Smarandache subgroupoid and A is Smarandache left ideal and not a Smarandache right ideal. Easy to verify. Theorem 9. Let G be a groupoid. An ideal of G in general is not a Smarandache ideal of G even if G is a Smarandache groupoid. Proof. By an example. Consider the groupoid G = {Z6 , ∗, (2, 4)} given by the following table. ∗ 0 1 2 3 4 5 0 0 4 2 0 4 2 1 2 0 4 2 0 4 2 4 2 0 4 2 0 3 0 4 2 0 4 2 4 2 0 4 2 0 4 5 4 2 0 4 2 0 Clearly G is a Smarandache groupoid for {0, 3} is a semigroup of G. Now, {0, 4, 2} is an ideal of G but is not a Smarandache ideal as {0, 4, 2} is not a Smarandache subgroupoid. Definition 10: Let G be a Smarandache groupoid and V be a Smarandache subgroupoid of G. We say V is a Smarandache semi-normal subgroupoid if: 1. aV = X for all a ∈ G; 2. V a = Y for all a ∈ G, where either X or Y is a Smarandache subgroupoid of G but X and Y are both subgroupoids. Example 6: Consider the groupoid G = {Z6 , ∗, (4, 5)} given by the table. ∗ 0 1 2 3 4 5 0 0 5 4 3 2 1 1 4 3 2 1 0 5 2 2 1 0 5 4 3 3 0 5 4 3 2 1 4 4 3 2 1 0 5 5 2 1 0 5 4 3 Vol. 1 31 Smarandache groupoids Clearly G is a Smarandache groupoid as {3} is a semigroup. Take A = {1, 3, 5}. A is also a Smarandache subgroupoid. Now aA = A is a Smarandache groupoid. Aa = {0, 2, 4}. {0, 2, 4} is not a Smarandache subgroupoid of G. Hence A is a Smarandache semi-normal subgroupoid. Definition 11: Let A be a Smarandache groupoid and V be a Smarandache subgroupoid. V is said to be Smarandache normal subgroupoid if aV = X and V a = Y where both X and Y are Smarandache subgroupoids of G. Theorem 12. Every Smarandache normal subgroupoid is a Smarandache semi-normal subgroupoid, and not conversely. Proof . By the definition 10 and 11, we see every Smarandache normal subgroupoid is Smarandache semi-normal subgroupoid. We prove the converse by an example. In example 6 we see A is a Smarandache semi-normal subgroupoid but not a normal subgroupoid as Aa = {0, 2, 4} is only a subgroupoid and not a Smarandache subgroupoid. Example 7: Let G = {Z8 , ∗, (2, 6)} be a groupoid given by the following table: ∗ 0 1 2 3 4 5 6 7 0 0 6 4 2 0 6 4 2 1 2 0 6 4 2 0 6 4 2 4 2 0 6 4 2 0 6 3 6 4 2 0 6 4 2 0 4 0 6 4 2 0 6 4 2 5 2 0 6 4 2 0 6 4 6 4 2 0 6 4 2 0 6 7 6 4 2 0 6 4 2 0 Clearly G is a Smarandache groupoid for {0, 4} is a semigroupoid G. A = {0, 2, 4, 6} is a Smarandache subgroupoid. Clearly Aa = A for all a ∈ G. So A is a Smarandache normal subgroupoid of G. Definition 13: Let G be a Smarandache groupoid H and P be subgroupoids of G, we say H and P are Smarandache semi-conjugate subgroupoids of G if: 1. H and P are Smarandache subgroupoids. 2. H = xP or P x, for some x ∈ G. 3. P = xH or Hx, for some x ∈ G. Example 8: Consider the groupoid G = {Z12 , ∗, (1, 3)} which is given by the following table: 32 W.B.Vasantha Kandasamy No. 2 ∗ 0 1 2 3 4 5 6 7 8 9 10 11 0 0 3 6 9 0 3 6 9 0 3 6 9 1 1 4 7 10 1 4 7 10 1 4 7 10 2 2 5 8 11 2 5 8 11 2 5 8 11 3 3 6 9 0 3 6 9 0 3 6 9 0 4 4 7 10 1 4 7 10 1 4 7 10 1 5 5 8 11 2 5 8 11 2 5 8 11 2 6 6 9 0 3 6 9 0 3 6 9 0 3 7 7 10 1 4 7 10 1 4 7 10 1 4 8 8 11 2 5 8 11 2 5 8 11 2 5 9 9 0 3 6 9 0 3 6 9 0 3 6 10 10 1 4 7 10 1 4 7 10 1 4 7 11 11 2 5 8 11 2 5 8 11 2 5 8 Clearly G is a Smarandache groupoid for {0, 6} is a semigroup of G. Let A1 = {0, 3, 6, 9} and A2 = {2, 5, 8, 11} be two subgroupoids. Clearly A1 and A2 are Smarandache subgroups of G as {0, 6} and {2, 8} are semigroups of A1 and A2 respectively. Now: A1 = 3{2, 5, 8, 11} = 3A2 = {0, 3, 6, 9} and similarly: A2 = 2{0, 3, 6, 9} = 2A1 . Hence A1 and A2 are conjugate Smarandache subgroupoids of G. Definition 15: Let G1 , G2 , G3 , · · · , Gn be n groupoids. We say G = G1 × G2 × · · · × Gn is a Smarandache direct product of groupoids if G has a proper subset H of G which is a semigroup under the operations of G. It is important to note that each Gi need not be a Smarandache groupoid for in that case G will be obviously a Smarandache groupoid. Hence we take any set of n groupoids and find the direct product. 0 Definition 16: Let (G, ∗) and (G , ◦) be any two Smarandache groupoids. A map φ 0 from (G, ∗) to (G , ◦) is said to be a Smarandache groupoid homomorphism if φ(a ∗ b) = φ(a) ◦ φ(b) for all a, b ∈ A. We say the Smarandache phism. groupoid homomorphism is an isomorphism if φ is an isomor- Definition 17: Let G be a Smarandache groupoid. We say G is a Smarandache commutative groupoid if there is a proper subset A of G which is a commutative semigroup under the operation of G. Vol. 1 33 Smarandache groupoids Definition 18: Let G be Smarandache groupoid. We say G is Smarandache inner commutative groupoid if every semigroup contained in every Smarandache subgroupoid of G is commutative. Theorem 19. Every Smarandache inner commutative groupoid G is a Smarandache commutative groupoid and not conversely. Proof. By the very definition 18 and 19 we see if G is a Smarandache inner commutative groupoid then G is Smarandache commutative groupoid. To prove the converse we prove it by an example. Let Z2 = {0, 1} be integers modulo 2. Consider set of all 2 × 2 matrices with entries from Z2 = (0, 1) denote it by M2×2 . M2×2       0 0 0 1    ,    0 0 0 0    =   0 1 1 0    ,    0 1 0 1       0 0 0 1 1 0 1 1 ,   ,   ,   ,   1 0 0 0 0 1 1 0   1 1 0 0 1 1 1 0 ,   ,  1 1 0 1 1 1 =  A◦B  =  a2 a3 a4 ◦ , ,   , a2 a3 a4   0 ,  1 0 0 0 ,◦ b2 b3 b4 1  0 1           0  .   1       1 0   b1 b2 b3 b4  in M2×2 .   a1 b3 + a2 b1 (mod2) a1 b4 + a2 b2 (mod2) a3 b3 + a4 b1 (mod2) a3 b4 + a4 b2 (mod2)    1   b1 Clearly (M2×2 , ◦) is a Smarandache groupoid for  0 1 1 1   and B =     0 So   0 1 1 0  a1   a1     0 0  M2×2 is made into a groupoid by for A =  ,  0 0  1 0 0 0   ◦  1 0 0 0  =  0 0 0 0 . is a semigroup.         0 0 1 0 0 1 1 , , , Now consider A1 =   0 0 0 0 0 0 0 dache groupoid but A1 is non-commutative Smarandache     groupoid   0 0 1 1 1 0 , , commutative semigroupoid S. S =   0 0 0 0 0 0    , ◦ is a Smaran 0 for A a non-  1 contains    1 0  , ◦ such that  ◦  0 0  1 34 W.B.Vasantha Kandasamy   1 1   0 0    1 1  No. 2   1 0 1 0 =  and  ◦ = . So (M2×2 , ◦) is a Smaran0 0 0 0 0 0 0 0 0 0 dache commutative groupoid but not Smarandache inner commutative groupoid. Definition 20: A groupoid G is said to be a M ouf ang groupoid if for every x, y, z in G we have (xy)(zx) = (x(yz))x. Definition 21: A Smarandache groupoid (G, ∗) is said to be Smarandache M ouf ang groupoid if there exists H ⊂ G such that H is a Smarandache groupoid satisfying the Moufang identity: (xy)(zx) = (x(yz)x) for all x, y, z in H. Definition 22: Let S be a Smarandache groupoid. If every Smarandache subgroupoid H of S satisfies the Moufang identity for all x, y, z in H then S is a Smarandache Strong M ouf ang groupoid. Theorem 23. Every Smarandache Strong Moufang groupoid is a Smarandache Moufang groupoidand not conversely. Proof . Every Strong Smarandache Moufang groupoid is a Smarandache Moufang groupoid. The proof of the converse can be proved by constructing examples. Definition 24: A groupoid G is said to be a Bol groupoid if ((xy)z)y = x((yz))y for all x, y, z ∈ G. Definition 25: Let G be a groupoid. G is said to be Smarandache − Bol groupoid if G has a subgroupoid H of G such that H is a Smarandache subgroupoid and satisfies the identity ((xy)z)y = x((yz))y for all x, y, z in H. Definition 26: Let G be a groupoid. We say G is Smarandache Strong Bol groupoid if every Smarandache subgroupoid of G is a Bol groupoid. Theorem 27. Every Smarandache Strong Bol groupoid is a Smarandache Bol groupoid and the converse is not true. Proof . Obvious. Theorem 28. Let Zn = {0, 1, 2, · · · , n − 1} be the set of integers modulo n. Let G = {Zn , ∗, (t, u)} be a Smarandache groupoid. G is a Smarandache Bol groupoid if t3 = t(mod n) and u2 = u(mod n). Proof . Easy to verify. Example 9: Let G = {Z6 , ∗, (2, 3)} defined by the following table:  ∗ 0 1 2 3 4 5 0 0 3 0 3 0 3 1 2 5 2 5 2 5 2 4 1 4 1 4 1 3 0 3 0 3 0 3 4 2 5 2 5 2 5 5 4 1 4 1 4 1 {0, 3} is a Smarandache subgroupoid and since 23 = 2(mod is a Smarandache Bol groupoid. 6) and 32 = 3(mod 6) we see G Vol. 1 Smarandache groupoids 35 Problem 2: Let {0, 1, 2, · · · , n − 1} be the ring of integers modulo n. G = {Zn , ∗, (t, u)} be a groupoid. Find conditions on n, t and u so that G: 1. is a Smarandache groupoid. 2. has Smarandache semi-normal subgroupoids. 3. has Smarandache normal subgroupoids. 4. is Smarandache commutative. 5. is Smarandache inner commutative. 6. is a Smarandache-Bol groupoid. 7. is a Smarandache Strong Bol groupoid. 8. is a Smarandache-Moufang groupoid. 9. is a Smarandache-Strong-Moufang groupoid. 10. has always a pair of Smarandache conjugate subgroupoid. References [1] R.H.Bruck, A Survey of Binary Systems, Springer Verlag, 1958. [2] Raul Padilla, Smarandache Algebraic Strucures, Bulletin of Pure and Applied Sciences, Delhi, Vol. 17E, 1(1998), 119-121. http://www.gallup.unm.edu/∼ Smarandache/ALG-S-TXT.TXT. [3] W.B.Vasantha Kandasamy, On ordered groupoids and its groupoid rings, J. of Mathematics and Comp. Sci., Vol. 9, 145-147, 1996. Scientia Magna Vol. 1 (2005), No. 2, 36-38 On the primitive numbers of power P and its mean value properties1 Ding Liping Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China,710069 Abstract Let p be a prime, n be any fixed positive integer. Sp (n) denotes the smallest positive integer such that Sp (n)! is divisible by pn . In this paper, we study the mean value properties of Sp (n) for p, and give a sharp asymptotic formula for it. Keywords Primitive numbers; Mean value; Asymptotic formula. §1. Introduction Let p be a prime, n be any fixed positive integer, Sp (n) denotes the smallest positive integer such that Sp (n)! is divisible by pn . For example, S3 (1) = 3, S3 (2) = 6, S3 (3) = 9, S3 (4) = 9, S3 (5) = 12, · · · . In problem 49 of book [1], Professor F. Smarandache asks us to study the properties of the sequence Sp (n). About this problem, some asymptotic properties of this sequence have been studied by Zhang Wenpeng and Liu Duansen [2], they proved that ¶ µ p Sp (n) = (p − 1)n + O log n . log p The problem is interesting because it can help us to calculate the Smarandache function. In this paper, we use the elementary methods to study the mean value properties of Sp (n) for p, and give a sharp asymptotic formula for it. That is, we shall prove the following: Theorem Let x ≥ 2, for any fixed positive integer n, we have the asymptotic formula X p≤x µ ¶ k−1 X nam x2 nx2 nx2 Sp (n) = + +O , 2 log x m=1 logm+1 x logk+1 x where am (m = 1, 2, · · · , k − 1) are computable constants. §2. Some Lemmas To complete the proof of the theorem, we need the following: Lemma Let p be a prime, n be any fixed positive integer. Then we have the estimate (p − 1)n ≤ Sp (n) ≤ np. 1 This work is supported by the N.S.F(10271093) and P.N.S.F of P.R.China Vol. 1 On the primitive numbers of power P 37 Proof. Let Sp (n) = k = a1 pα1 + a2 pα2 + · · · + as pαs with αs > αs−1 > · · · > α1 ≥ 0 under the base p. Then from the definition of Sp (n) we know that p | Sp (n)! and the Sp (n) denotes the smallest integer satisfy the condition. However, let (np)! = 1 · 2 · 3 · · · p · (p + 1) · · · 2p · (2p + 1) · · · np = upl . where l ≥ n, p†u. So combining this and p | Sp (n)! we can easily obtain Sp (n) ≤ np. (1) On the other hand, from the definition of Sp (n) we know that p | Sp (n)! and pn † (Sp (n) − 1)!, so that α1 ≥ 1, note that the factorization of Sp (n)! into prime powr is Y k! = q αq (k) . q≤k where Q denotes the product over all prime , and q≤k αq (k) = ∞ X k [ i] q i=1 because p | Sp (n)!, so we have n ≤ αp (k) = ∞ X k k [ i] = p p − 1 i=1 or (p − 1)n ≤ k (2) combining (1) and (2) we immediately get the estimate (p − 1)n ≤ Sp (n) ≤ np. This completes the proof of the lemma. §3. Proof of the theorem In this section, we complete the proof of Theorem. Based on the result of lemma (p − 1)n ≤ Sp (n) ≤ np we can easily get X (p − 1)n ≤ p≤x Let   1, a(n) =  0, X p≤x Sp (n) ≤ X p≤x if n is prime; otherwise, np. 38 Ding Liping No. 2 Then from [3] we know that for any positive integer k, X a(n) = π(x) = n≤x µ ¶ k−1 X m! x x (1 + ) + O . logx logm x logk+1 x m=1 By Abel’s identity we have X X p= a(m)m p≤x m≤x Z x = π(x)x − π(t)dt 2 µ ¶ Z x k−1 k−1 X m! x2 X m! t x2 x2 + − (1 + )dt + O logx logx m=1 logm x logm t logk+1 x 2 logt m=1 µ ¶ k−1 X x2 am x2 x2 + + O 2 log x m=1 log( m + 1)x logk+1 x = = where am (m = 1, 2, · · · , k − 1) are computable constants. From above we have X (p − 1) = p≤x X p≤x µ ¶ k−1 X x2 am x2 x2 + p − π(x) = +O . 2 log x m=1 log( m + 1)x logk+1 x Therefore X p≤x Sp (n) = X p≤x k == ¶ µ k−1 X x2 am x2 x2 + . + O 2 log x m=1 log( m + 1)x logk+1 x This completes the proof of the theorem. Acknowledgments The author express his gratitude to his supervisor Professor Zhang Wenpeng for his very helpful and detailed instructions. References [1] F.Smarandache, Only problems, Not Solutions, Chicago, Xiquan Publ. House, 1993. [2] Zhang Wenpeng and Liu Duansen, On the primitive numbers of power P and its asymptotic property, Smarandache Notions Journal, 13(2002), 173-175. [3] M.Ram Murty, Problems in analytic number theory, Springer-Verlag, New York, 2001, pp. 36 Scientia Magna Vol. 1 (2005), No. 2, 39-45 Consecutive, reversed, mirror, and symmetric Smarandache sequence of triangular numbers Delfim F.M.Torres† and Viorica Teca ‡ † Department of Mathematics University of Aveiro,Portugal email: delfim@mat.ua.pt ‡ Faculty of Mathematics-Informatics University of Craiova,Romania email: viorica teca@yahoo.com Abstract We use the Maple system to check the investigations of S.S.Gupta regarding the Smarandache consecutive and the reversed Smarandache sequence of triangular numbers [Smarandache Notions Journal,Vol. 14, 2004, pp.366-368]. Furthermore, we extend previous investigations to the mirror and symmetric Smarandache sequences of triangular numbers. Pn The nth triangular number tn , n ∈ N , is defined by tn = i=1 i = n(n + 1)/2. These numbers were first studied by the Pythagoreans. The first k terms of the triangular sequence {tn }∞ n=1 are easily obtained in Maple: > t:=n − >n∗(n+1)/2: > first :=k − > seq(t(n), n=1,· · · ,k): For example: > first(20); 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210 In this short note we are interested in studying Smarandache sequences of triangular numbers with the help of the Maple system. To define the Smarandache sequences, it is convenient to introduce first the concatenation operation. Given two positive integer numbers n and m, the concatenation operation conc is defined in Maple by the following function: > conc :=(n, m) − > n∗10length(m) +m: For example, > conc(12, 345); 12345 Given a positive integer sequence {un }∞ n=1 , we define the corresponding Smarandache Consecutive Sequence {scsn }∞ recursively: n=1 scs1 = u1 , 40 Delfim F.M.Torres and Viorica Teca No. 2 scsn = conc(scsn−1 , un ). In Maple we define: > scs n :=(u, n) − > if n = 1 then u(1) else conc(scs n(u, n−1), u(n)) fi: > scs := (u, n) − > seq(scs n(u, i), i=1· · · n): The standard Smarandache consecutive sequence, introduced by the Romanian mathematician Florentin Smarandache, is obtained when one chooses un = n, ∀n ∈ N . The first 10 terms are: > scs(n->n,10); 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 12345678910 Another example of a Smarandache consecutive sequence is the Smarandache consecutive sequence of triangular numbers. With our Maple definitions, the first 10 terms of such sequence are obtained with the following command: > scs(t, 10); 1, 13, 136, 13610, 1361015, 136101521, 13610152128, 1361015212836, 136101521283645, 13610152128364555, Sometimes, it is preferred to display Smarandache sequence in ”triangular form”. > show :=L − >map(i − >print(i), L): > show([scs(t, 10)]): 1 13 136 13610 1361015 136101521 13610152128 1361015212836 136101521283645 13610152128364555 The Reversed Smarandache Sequence (rss) associated with a given sequence {un }∞ n=1 , is defined recursively by rss1 = u1 rssn = conc(un , rssn−1 ). In Maple we propose the following definitions: > rss n :=(u, n) − > if n = 1 then u(1) else conc(u(n), res n(u, n−1)) fi: > rss :=(u, n) − > seq(rss n(u, i), i=1,· · · n): Vol. 1 Smarandache sequence of triangular numbers 41 The first terms of the reversed Smarandache sequence of triangular numbers are now easily obtained: > rss(t, 10); 1, 31, 631, 10631, 1510631, 211510631, 28211510631, 3628211510631, 453628211510631, 55453628211510631, We define the Smarandche Mirror Sequence (sms) as follows: sms1 = u1 , smsn = conc(conc(un , smsn−1 ), un ) > sms n :=(u, n) − > if n = 1 then > u(1) > else > conc(conc(u(n), sms n(u, n−1)), u(n)) > sms :=(u, n) − >seq(sms n(u, i), i=1· · · n): The first 10 terms of the Smarandache mirror sequence introduced by Smarandache are: > sms(n − >, 10); 1, 212, 32123, 4321234, 543212345, 65432123456, 7654321234567, 876543212345678, 98765432123456789, 109876543212345678910 We are interested in the Smarandache mirror sequence of triangular numbers. The first 10 terms are: > sms(t, 10); 1, 313, 63136, 106313610, 1510631361015, 21151063136101521, 282115106313610152128, 3628211510631361015212836, 45362821151063136101521283645, 554536282115106313610152128364555, Finally, we define the Smarandache Symmetric Sequence (sss). For that we introduce the function ”But Last Digit” (bld): > bld :=n − > iquo(n,10): > bld(123); 12 If the integer number is a one-digit number, then function bld returns zero: > bld(3); 0 This is important: with our conc function, the concatenation of zero with a positive integer n gives n. > conc(bld(1), 3); 3 42 Delfim F.M.Torres and Viorica Teca No. 2 The Smarandache Symmetric Sequence (sss) is now easily defined, appealing to the Smarandache consecutive, and reversed Smarandache sequences: sss2n−1 = conc(bld(scs2n−1 ), rss2n−1 ), sss2n = conc(scs2n , rss2n ), n ∈ N . In Maple, we give the following definitions: > sss n :=(u, n) − > if type(n, odd) then > conc(bld(scs n(u, (n+1)/2), rss (u, (n+1)/2)) > else > conc(scs n(u, n/2), rss n(u, n/2)) > fi: > sss :=(u, n) − > seq(sss n(u, i), i=1· · · n): The first terms of Smarandache’s symmetric sequence are > sss(n − > n, 10); 1, 11, 121, 1221, 12321, 123321, 1234321, 12344321, 123454321, 1234554321 while the first 10 terms of the Smarandache symmetric sequence of triangular numbers are > sss(t, 10); 1, 11, 131, 1331, 13631, 136631, 136110631, 1361010631, 1361011510631, 13610151510631, One interesting question is to find prime numbers in the above defined Smarandache sequences of triangular numbers. We will restrict our search to the first 1000 terms of each sequence. All computations were done with Maple 9 runing on a 2.00Ghz Pentium 4 with 256Mb RAM. We begin by collecting four lists with the first 1000 terms of the consecutive, reversed, mirror, and symmetric Smarandache sequences of triangular numbers: > st:=time(): Lscs1000:=[scs(t, 1000)]: print(”%a seconds”, round(time()-st)); 20 seconds > st:=time(): Lrss1000:=[rss(t, 1000)]: print(”%a seconds”, round(time()-st)); 75 seconds > st:=time(): Lsms1000:=[sms(t, 1000)]: print(”%a seconds”, round(time()-st)); 212 seconds > st:=time(): Lsss1000:=[sss(t, 1000)]: print(”%a seconds”, round(time()-st)); 26 seconds We note that scs1000 and rss1000 are positive integer numbers with 5354 digits; > length(Lscs1000[1000]), length(Lrss1000[1000]); 5354, 5354 while sms1000 and sss1000 have, respectively, 10707 and 4708 digits. > length(Lsms1000[1000]), length(Lsss1000[1000]); 10707, 4708 Vol. 1 Smarandache sequence of triangular numbers 43 There exist two primes (13 and 136101521) among the first 1000 terms of the Smarandache consecutive sequence of triangular numbers; > st := time() > select(isprime, Lscs1000); > printf(”%a minutes”, round((time()-st)/60)); [13, 136101521] 9 minutes six primes among the first 1000 terms of the reversed Smarandache sequence of triangular numbers; > st := time() > select(isprime, Lrss1000); > printf(”%a minutes”, round((time()-st)/60)); [31, 631, 10631, 55453628211510631, 786655453628211510631, 10591786655453628211510631] 31 minutes only one prime (313) among the first 600 terms of the Smarandache mirror sequence of triangular numbers; > length(Lsms1000[600]); # sms {600} is a number with 5907 digits 5907 > st := time() > select(isprime, Lsms1000[1· · · 600]); > printf(”%a minutes”, round((time()-st)/60)); [313] 3 minutes and five primes among the first 1000 terms of the Smarandache symmetric sequence of triangular numbers (the fifth prime is an integer with 336 digits). > st := time() > select(isprime, Lsss1000); > printf(”%a minutes”, round((time()-st)/60)); [11,131,136110631,1361015212836455566789110512012010591786655453628211510631, 1361015212836455566789110512013615317119021023125327630032535137840643546549652856159 5630666703741780820861903946990103510811128117612251275132613781431148515401596165316 5315961540148514311378132612751225117611281081103599094690386182078074170366663059556 152849646543540637835132530027625323121019017115313612010591786655453628211510631] 19 minutes > length(%[5]); 336 How many primes are there in the above defined Smarandache sequences of triangular numbers? This seems to be an open question. Another interesting question is to find triangular 44 Delfim F.M.Torres and Viorica Teca No. 2 numbers in the Smarandache sequences of triangular numbers. We begin by defining in Maple the boolean function istriangular. > istriangular := n − > evalb(nops(select(i− >evalb(whattype(i)=integer),)[solve(t(k)=n)])> 0) There exist two triangular numbers (1 and 136) among the first 1000 terms of the Smarandache consecutive sequence of triangular numbers; > st := time() > select(istriangular, Lscs1000); > printf(”%a seconds”, round(time()-st)); [1, 136] 6 seconds while the other Smarandache sequences of triangular numbers only show, among the first 1000 terms, the trival triangular number 1: > st := time() > select(istriangular, Lrss1000); > printf(”%a seconds”, round(time()-st)); [1] 6 seconds > st := time() > select(istriangular, Lsms1000); > printf(”%a seconds”, round(time()-st)); [1] 10 seconds > st := time() > select(istriangular, Lsss1000); > printf(”%a seconds”, round(time()-st)); [1] 6 seconds Does exist more triangular numbers in the Smarandache sequences of triangular numbers? This is, to the best of our knowledge, an open question needing further investigations. Since checking if a number is triangular is much faster than to check if a number if prime, we invite the readers to continue our research of triangular numbers for besides the 1000th term of the Smarandache sequences of triangular numbers. We look forward to readers discoveries. Vol. 1 Smarandache sequence of triangular numbers 45 References [1] Shyam Sunder Gupta, Smarandache Sequence of Triangular Numbers, Smarandache Notions Journal, 14(2004), 366-368 . Scientia Magna Vol. 1 (2005), No. 2, 46-48 On the square-free number sequence Ren Dongmei Research Center for Basic Science, Xi’an Jiaotong University Xi’an, Shaanxi, P.R.China Abstract The main purpose of this paper is to study the number of the square-free number sequence, and give two interesting asymptotic formulas for it. At last, give another asymptotic formula and a corollary. Keywords Square-free number sequence; Asymptotic formula. §1. Introduction A number is called a square-free number if its digits don’t contain the numbers: 0, 1, 4, 9. Let A denote the set of all square-free numbers. In reference [1], Professor F. Smarandache asked us to study the properties of the square-free number sequence. About this problem, it seems that none had studied it, at least we have not seen such a paper before. In this paper, we use the elementary method to study the number of the square-free X number sequence, and obtain two interesting asymptotic formulas for it. That is, let S(x) = 1, we shall prove n≤x,n∈A the followings: Theorem 1. For any real number x ≥ 1, we have the asymptotic formula ln S(x) = ln 6 × ln x + O(1). ln 10 Theorem 2. For any real number x ≥ 1, we have the asymptotic formula X ³ 2 ln 2 ´ 1 = x + O x ln 10 , n≤x,n∈B where B denote the complementary set of those numbers whose all digits are square numbers. Let B 0 denote the set of those numbers whose all digits are square numbers. Then we have the following: Theorem 3. For any real number x ≥ 1,we have the asymptotic formula X n≤x,n∈B ¶ µ ln 5 1 2 = ln x + γ − C + O x− ln 10 , n where C is a computable constant, γ denotes the Euler’s constant. Let A0 denote the complementary set of A, we have following: Corollary. For any real number x ≥ 1, we have the asymptotic formula Vol. 1 On the Square-Free number sequence 47 µ ¶ ln 5 1 3 − ln 10 = ln x + γ − D + O x , n X n≤x,n∈A0 where D is a computable constant. §2. Proof of Theorems In this section, we shall complete the proof of Theorems. First we need the following one simple lemma. Lemma. For any real number x ≥ 1, we have the asymptotic formula X n≤x,n∈B0 µ ¶ ln 5 1 2 − ln 10 =C +O x . n Proof. In the interval [10r−1 , 10r ), (r ≥ 2), there are 3 × 4r−1 numbers belong to B 0 , and 1 every number’s reciprocal isn’t greater than 10r−1 ; when r = 1, there are 4 numbers belong to 0 B and their reciprocals aren’t greater than 1. Then we have then ∞ X 1 X 4r <3+ 3× r, n 10 0 r=1 n∈B P n∈B0 1 is convergent to a constant C. So X n≤x,n∈B0 X 1 1 = − n n 0 n∈B X n>x,n∈B0 Ã∞ ! µ ¶ X 3 × 4r ln 5 1 2 − ln 10 =C +O =C +O x . n 10r r=k Now we come to prove Theorem 1. First for any real number x ≥ 1, there exists a nonnegative integer k, such that 10k ≤ x < 10k+1 (k ≥ 1) therefore k ≤ log x < k + 1. If a number belongs to A, then its digits only contain these six numbers: 2, 3, 5, 6, 7, 8. So in the interval [10r−1 , 10r )(r ≥ 1), there are 6r numbers belong to A. Then we have X 1≤ n≤x,n∈A k+1 X 6r = r=1 ln 6 6 6k+2 62 × (6k+1 − 1) < < × x ln 10 , 5 5 5 and X n≤x,n∈A 1≥ k X 6r = r=1 ln 6 1 6 × (6k − 1) ≥ 6k > × x ln 10 . 5 6 So we have ln 6 1 × x ln 10 < 6 X n≤x,n∈A 1< ln 6 62 × x ln 10 . 5 Taking the logarithm computation on both sides of the above, we get 48 Ren Dongmei ³ ln 6 ´ 1 < ln x ln 10 + (2 × ln 6 − ln 5). X ln 6 ln(x ln 10 ) + (− ln 6) < No. 2 n≤x,n∈A So  ln S(x) = ln   X n≤x,n∈A ³ ln 6 ´ ln 6 1 = ln x ln 10 + O(1) = × ln x + O(1). ln 10 This proves the Theorem 1. Now we prove Theorem 2. It is clear that if a number doesn’t belong to B, then all of its digits are square numbers. So in the interval [10r−1 , 10r ), (r ≥ 2), there are 3 × 4r−1 numbers don’t belong to B; when r = 1, there are 4 numbers don’t belong to B. Then we have X 1= n≤x,n∈B X n≤x 1− X 1 n≤x,n∈B0 ¡ ¢ = x + O 4 + 3 × 4 + 3 × 42 + · · · + 3 × 4k ³ 2×ln 2 ´ ¡ ¢ = x + O 4k+1 = x + O x ln 10 . This completes the proof of the Theorem 2. Now we prove the Theorem 3. In reference [2], we know the asymptotic formula: µ ¶ X 1 1 = ln x + γ + O , n x n≤x where γ is the Euler’s constant. Then from this asymptotic formula and the above Lemma, we have X X 1 X 1 1 = − n n n n≤x,n∈B n≤x n≤x,n∈B0 µ ¶ µ ¶ ln 5 1 2 − ln 10 = ln x + γ + O −C +O x x µ ¶ ln 5 2 − ln 10 = ln x + γ − C + O x . This completes the proof of the Theorem 3. Now the Corollary immediately follows from the Lemma and Theorem 3. Reference [1] F.Smarandache, Only problems, Not Solutions, Xiquan Publ. House, Chicago, 1993. [2] Tom M.Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. Scientia Magna Vol. 1 (2005), No. 2, 49-51 On finite Smarandache near-rings T.Ramaraj † and N.Kannappa ‡ Lecture in Selection Grade Department of Mathematics T.B.M.L.College Porayar-609307 Nagappattinam-(Dt), Tamil Nadu, India Reader in Mathematics A.V.V.M.Pushpam College (Autonomous) Poondi-613 503 Tanjore-(Dt), Tamil Nadu, India Abstract In this paper we study the Finite Smarandache-2-algebraic structure of Finite-near-ring, namely, Finite-Smarandache-near-ring, written as Finite-S-near-ring. We define Finite Smarandache near-ring with examples. We introduce some equivalent conditions for Finite S-near-ring and obtain some of its properties. Keywords Finite-S-near-ring; Finite-Smarandache-near-ring. §1. Introduction In this paper, we studied Finite-Smarandache 2-algebraic structure of Finite-near-rings, namely, Finite-Smarandache-near-ring, written as Finite-S-near-ring. A Finite-Smarandache 2-algebraic structure on a Finite-set N means a weak algebraic structure A0 on N such that there exist a proper subset M of N , which is embedded with a stronger algebraic structure A1 , stronger algebraic structure means satisfying more axioms, by proper subset means a subset different from the empty set, from the unit element if any, from the whole set [5]. By a Finitenear-ring N , we mean a zero-symmetric Finite- right-near-ring. For basic concept of near-ring we refer to Gunter Pilz [2]. Definition 1. A Finite-near-ring N is said to be Finite-Smarandache-near-ring. If a proper subset M of N is a Finite-near-field under the same induced operations in N . Example 1 [2]. Let N = {0, n1 , n2 , n3 } be the Finite-near-ring defined by: Let M = {0, n1 } ⊂ N be a Finite-near-field. Defined by Now (N, +, .) is a Finite-S-near-ring . Example 2 [4]. Let N = {0, 6, 12, 18, 24, 30, 36, 42, 48, 54} (mod 60) be the Finitenear-ring since every ring is a near-ring. Now N is a Finite-near-ring, Whose proper subset M = {0, 12, 24, 36, 48} (mod 60) is a Finite-field. Since every field is a near-field, then M is a Finite-near-field. Therefore N is a Finite-S-near-ring. Theorem 1. Let N be a Finite-near-ring. N is a Finite-S-near-ring if and only if there exist a proper subset M of N , either M ∼ = Mc (z2 ) or Zp , integers modulo p, a prime number. Proof. Part-I: We assume that N is a Finite-S-near-ring. By definition, there exist a proper subset M of N is a Finite-near-field. By Gunter Pilz Theorem (8.1)[2], either M ∼ = 50 T.Ramaraj No. 2 Mc (z2 ) or zero-symmetric. Since Zp,S is zero-symmetric and Finite-fields implies Zp , S are zerosymmetric and Finite-near-fields because every field is a near-field. Therefore in particular M is Zp . Part-II: We assume that a proper subset M of N , either M ∼ = Mc (z2 ) or Zp . Since Mc (z2 ) and Zp are Finite-near-fields. Then M is a Finite-near-field. By definition, N is a Finite-S-near-ring. Theorem. Let N be a Finite-near-ring. N is a Finite-S-near-ring if and only if there exist a proper subset M of N such that every element in M satisfying the polynomial xpm − x. Proof. Part-I: We assume that N is a Finite-S-near-ring. By definition, there exist a proper subset M of N is a Finite-near-field. By Gunter Pilz, Theorem (8.13)[2]. If M is a Finite-near-field, then there exist p ∈ P, ∃m ∈ M such that | M |= pm . According to I.N.Herstein[3]. If the Finite-near-field M has pm element, then every a ∈ M satisfies apm = a, since every field is a near-field. Now M is a Finite-near-field having pm element, every element a in M satisfies apm = a. Therefore every element in M satisfying the polynomial xpm − x. Part-II: We assume that there exist a proper subset M of N such that every element in M satisfying the polynomial xpm − x, which implies M has pm element. According to I.N.Herstein[3], For every prime number p and every positive integer m, there is a unique field having pm element. Hence M is a Finite-field implies M is a Finite-near-field. By definition, N is a Finite-S-near-ring. Theorem 3. Let N be a Finite-near-ring. N is a Finite-S-near-ring if and only if M has no proper left ideals and M0 6= M . Where M is a proper sub near-ring of N , in which idempotent commute and for each x ∈ M , there exist y ∈ M such that yx 6= 0. Proof. Part-I :We assume that N is a Finite-S-near-ring. By definition A proper subset M of N is a Finite-near-field. In [1] Theorem (4),it is zero-symmetric and hence every left-ideal is a M-subgroup. Let M1 6= 0 be a M-subgroup and m1 6= 0 ∈ M1 . Then m−1 1 m1 = 1 ∈ M1 . therefore M = M1 . Hence M has no proper M-subgroup, which implies M has no proper left ideal. Part-II: We assume that a proper sub-near-ring M of N has no proper left ideals and M0 6= M , in which idempotent commute and for each x ∈ M there exist y ∈ M such that yx 6= 0. Let x 6= 0 in M . Let F (x) = {m ∈ M | mx = 0}. Clearly F (x) is a left ideal. Since there exist y ∈ M such that yx 6= 0. Then y ∈ / F (x). Hence F (x) = 0. Let φ : (M, +) −→ (M x, +) given by φ(m) = mx. Then φ is an isomorphism. Since M is finite then M x = M . Now by a theroem(2) in [1], M is a Finite-near-field. Therefore, by definition N is a Finite-S-near-ring. We summarize what has been studied in Theorem 4. Let N be a Finite-near-ring. Then the following conditions are equivalent. 1. A proper subset M of N , either M ∼ = Mc (z2 ) or Zp , integers modulo p, a prime number. 2. A proper subset M of N such that every element in M satisfying the polynomial xpm −x. 3. M has no proper left ideals and M0 6= M . Where M is a proper sub near-ring of N , in which idempotent commute and for each x ∈ M , there exist y ∈ M such that yx 6= 0. Theorem 5. Let N be a Finite-near-ring. If a proper subset M , sub near-ring of N , in which M has left identity and M is 0-primitive on M M . Then N is a Finite-S-near-ring. Proof. By Theorem(8.3)[2], the following conditions are equivalent: Vol. 1 On finite Smarandache near-rings 51 (1) M is a Finite-near-field; (2) M has left identity and M is 0-primitive on M M . Now Theorem is immediate. Theorem 6. Let N be a Finite-near-ring. If a proper subset M , sub near-ring of N , in which M has left identity and M is simple. Then N is a Finite-S-near-ring. Proof. By Theorem(8.3)[2], the following conditions are equivalent: (1) M is a Finite-near-field; (2) M has left identity and M is simple. Now the Theorem is immediate. Theorem 7. Let N be a Finite-near-ring. If a proper subset M , sub near-ring of N is a Finite-near-domain, then N is a Finite-S-near-ring. Proof. By Theorem(8.43)[2], a Finite-near- domain is a Finite-near-field. Therefore M is a Finite-near-field. By definition N is a Finite-S-near-ring. Theorem 8. Let N be a Finite-near-ring. If a proper subset M of N is a Finite-Integerdomain. Then N is a Finite-S-near-ring. Proof. By I.N.Herstein[3], every Finite-Integer-domain is a field, since every field is a near-field. Now M is a Finite-near-field. By definition N is a Finite-S-near-ring. Theorem 9. Let N be a Finite-near-ring. If a proper subset M of N is a Finite-divisionring. Then N is a Finite-S-near-ring. Proof. By Wedderburn’s Theorem(7.2.1)[3], a Finite-division-ring is a necessarily commutative field, which gives M is a field, implies M is a Finite-near-field. By definition N is a Finite-S-near-ring. References [1] P.Dheena, On near-field, J. Pure. Appl. Math., 17(3) (1986), 332-326. [2] G.Pilz, Near-ring, North Holland, Amsterdam, 1997. [3] I.N.herstein, Topics in algebra, Wiley Eastern Limited, New Delhi, 1993 [4] R.Padilla, Smarandache algebraic structures,presented to the Universidade do Minho, Baraga, Portugal, 18-23, June, 1999. [5] PlanetMath, Smarandache n-structure. Scientia Magna Vol. 1 (2005), No. 2, 52-54 Some interesting properties of the Smarandache function Kang Xiaoyu Editorial Board of Journal of Northwest University Xi’an, Shaanxi, P.R.China Abstract The main purpose of this paper is using the elementary method to study the property of the Smarandache function, and give an interesting result. Keywords Smarandache function; Additive property; Greatest prime divisor. §1. Introduction and results Let n be an positive integer, the famous Smarandache function S(n) is defined as following: S(n) = min{m : m ∈ N, n|m!}. About this function and many other Smarandache type function, many scholars have studied its properties, see [1], [2], [3] and [4]. Let p(n) denotes the greatest prime divisor of n, it is clear that S(n) ≥ p(n). In fact, S(n) = p(n) for almost all n, as noted by Erdös [5]. This means that the number of n ≤ x for which S(n) 6= p(n), denoted by N (x), is o(x). It is easily to show that S(p) = p and S(n) < n except for the case n = 4, n = p. So there have a closely relationship between S(n) and π(x): ¸ [x] · X S(n) π(x) = −1 + , n n=2 where π(x) denotes the number of primes up to x, and [x] denotes the greatest integer less than or equal to x. For two integer m and n, can you say S(mn) = S(m) + S(n) is true or false? It is difficult to say. For some m an n, it is true, but for some other numbers it is false. About this problem, J.Sandor [7] proved an very important conclusion. That is, for any positive integer k and any positive integers m1 , m2 , · · · , mk , we have the inequality à k ! k Y X S mi ≤ S(mi ). i=1 i=1 This paper as a note of [7], we shall prove the following two conclusions: Theorem 1. inequality For any integer k ≥ 2 and positive integers m1 , m2 , · · · , mk , we have the à S k Y i=1 ! mi ≤ k Y i=1 S(mi ). Vol. 1 Some interesting properties of the Smarandache function 53 Theorem 2. For any integer k ≥ 2, we can find infinite group numbers m1 , m2 , · · · , mk such that: à k ! k Y X S mi = S(mi ). i=1 i=1 §2. Proof of the theorems In this section, we will complete the proof of the Theorems. First we prove a special case of Theorem 1. That is, for any positive integers m and n, we have S(m)S(n) ≥ S(mn). If m = 1 ( or n = 1), then it is clear that S(m)S(n) ≥ S(mn). Now we suppose m ≥ 2 and n ≥ 2, so that S(m) ≥ 2, S(n) ≥ 2, mn ≥ m + n and S(m)S(n) ≥ S(m) + S(n). Note that m|S(m)!, n|S(n)!, we have mn|S(m)!S(n)!|((S(m) + S(n))!. Because S(m)S(n) ≥ S(m) + S(n), we have (S(m) + S(n))!|(S(m)S(n))!. That is, mn|S(m)!S(n)!|(S(m) + S(n))!|(S(m)S(n))!. From the definition of S(n) we may immediately deduce that S(mn) ≤ S(m)S(n). Now the theorem 1 follows from S(mn) ≤ S(m)S(n) and the mathematical induction. Proof of Theorem 2. For any integer n and prime p, if pα kn!, then we have α= ¸ ∞ · X n j=1 pj . Let ni are positive integers such that ni 6= nj , if i 6= j, where 1 ≤ i, j ≤ k, k ≥ 2 is any positive integer. Since ∞ · ni ¸ X p pn i − 1 ni −1 ni −2 = p + p + · · · + 1 = . pr p−1 r=1 For convenient, we let ui = pni −1 p−1 . So we have S(pui ) = pni , In general, we also have  k X i = 1, 2, · · · , k. (1)   pn i  ∞  k k  X X pni − 1 X  i=1  = = ui .    pr  p−1 r=1  i=1  i=1 So k ¡ ¢ X S pu1 +u2 +···+uk = pni . i=1 (2) 54 Kang Xiaoyu No. 2 Combining (1) and (2) we may immediately obtain à k ! k Y X ui S p = S(pui ). i=1 i=1 Let mi = pui , noting that there are infinity primes p and ni , we can easily get Theorem 2. This completes the proof of the theorems. References [1] C.Ashbacher, Some Properties of the Smarandache-Kurepa and Smarandache-Wagstaff Functions. Mathematics and Informatics Quarterly, 7(1997), 114-116. [2] A.Begay, Smarandache Ceil Functions Bulletin of Pure and Applied Sciences, 16(1997), 227-229. [3] Mark Farris and Patrick Mitchell, Bounding the Smarandache function Smarandache Notions Journal,13(2002), 37-42. [4] Kevin Ford, The normal behavior of the Smarandache function, Smarandache Notions Journal, 10(1999), 81-86. [5] P.Erdös, Problem 6674 Amer. Math. Monthly, 98(1991), 965. [6] Pan Chengdong and Pan Chengbiao, Element of the analytic number theory, Science Press, Beijing, (1991). [7] J.Sandor, On a inequality for the Smarandache function, Smarandache Notions Journal, 10(1999), 125-127. Scientia Magna Vol. 1 (2005), No. 2, 55-73 On automorphism groups of maps, surfaces and Smarandache geometries 1 Linfan Mao Institute of Systems, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, P.R.China Abstract A combinatorial map is a connected topological graph cellularly embedded in a surface. This report concentrates on the automorphism group of a map, which is related to the automorphism groups of a Klein surface and a Smarandache manifold, also applied to the enumeration of unrooted maps on orientable and non-orientable surfaces. A number of results for the automorphism groups of maps, Klein surfaces and Smarandache manifolds and the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are discovered. An classification for the closed s-manifolds by maps is found. Open problems related the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries are also presented in this report for the further applications of the combinatorial maps to the classical mathematics. Keywords Automorphism group; Surface; Map; Smarandache geometries; Map geometries; Classification. Part I. Terminology and Notations §1.1 Klein Surfaces P A Klein surface is a Hausdorff, connected, topological space S with a family = {(Ui , φi )|i ∈ I} such that the chart {Ui |i ∈ I} is an open covering of S, each map φi : Ui −→ Ai is a homeomorphism onto an open subset Ai of C or C + = {z ∈ C : Imz ≥ 0} and the transition functions φij = φi φ− j : φj (Ui \ Uj ) −→ φi (Ui \ Uj ). are dianalytic, where a mapping f : A −→ C is said dianalytic if equation) or ∂f ∂z = 0. ∂f ∂z = 0 (Cauchy-Riemann §1.2 {Riemann Surfaces}⊂ {Klein surfaces} §1.3 Embedding and Combinatorial Maps Embedding of a graph: For any connected graph Γ = (V (Γ), E(Γ)) and a surface S, an embedding of the graph Γ in the surface S is geometrical defined to be a continuous 1 − 1 mapping τ : Γ → S. The image 1 Reported at the Academy of Mathematics and Systems of Chinese Academy of Sciences. 56 Linfan Mao No. 2 τ (Γ) is contained in the 1-skeleton of a triangulation of the surface S. If each component in S − τ (Γ) homeomorphic to an open disk, then the embedding is an embedding. Map: A combinatorial map is a connected topological graph cellularly embedded in a surface. The Algebraic Definition of Maps: A combinatorial map M = (Xα,β , P) is defined to be a basic permutation P, i.e, for any x ∈ Xα,β , no integer k exists such that P k x = αx, acting on Xα,β , the disjoint union of quadricells Kx of x ∈ X (the base set), where K = {1, α, β, αβ} is the Klein group, satisfying the following two conditions: (Ci) αP = P −1 α; (Cii) the group ΨJ =< α, β, P > is transitive on Xα,β . §1.4 Orientation If the group ΨI =< αβ, P > is transitive on Xα,β , then M is non-orientable. Otherwise, orientable. §1.5 An Example of Maps K4 on the torus. Fig.1 ¸ (Xα,β , P): Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy, βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw} Vol. 1 On automorphisms groups of maps, surfaces and Smarandache geometries P 57 = (x, y, z)(αβx, u, w)(αβz, αβu, v) × (αβy, αβv, αβw)(αx, αz, αy)(βx, αw, αu) × (βz, αv, βu)(βy, βw, βv) Vertices: v1 = {(x, y, z), (αx, αz, αy)} v2 = {(αβx, u, w), (βx, αw, αu)} v3 = {(αβz, αβu, v), (βz, αv, βu)} v4 = {(αβy, αβv, αβw), (βy, βw, βv)} Edges: {e, αe, βe, αβe}, e ∈ {x, y, z, u, v, w} Faces: f1 = {(x, u, v, αβw, αβx, y, αβv, αβz), (βx, αz, αv, βy, αx, αw, βv, βu)} f2 = {(z, αβu, w, αβy), (βz, αy, βw, αu)} §1.6 Isomorphism of Maps 1 2 Two maps M1 = (Xα,β , P1 ) and M2 = (Xα,β , P2 ) are said to be isomorphic if there exists a bijection ξ 1 2 ξ : Xα,β −→ Xα,β 1 such that for ∀x ∈ Xα,β , ξα(x) = αξ(x), ξβ(x) = βξ(x), ξP1 (x) = P2 ξ(x). §1.7 Equivalence Two maps M1 , M2 underlying graph Γ are equivalent if there exists an isomorphism ζ between them induced by an element ξ, ξ ∈ AutΓ. Call ζ an equivalence between M1 , M2 . Certainly, on an orientable surface, an equivalence preserve the orientation on this surface. Theorem 1.1. Let M = (Xα,β , P) be a map with an underlying graph Γ, ∀g ∈ AutΓ. Then the extend action of g on Xα,β with X = E(Γ) is an automorphism of map M iff ∀v ∈ V (M ), g preserves the cyclic order of v. §1.8 Covering of Maps f = (X e f ] For two maps M α,β , P) and M = (Xα,β , P), call the map M covering the map M if ] ] there is a mapping π : X α,β → Xα,β such that ∀x ∈ Xα,β , e απ(x) = πα(x), βπ(x) = πβ(x), π P(x) = Pπ(x). ] Theorem 1.2. Let π : X α,β → Xα,β be a covering mapping. Then π is an isomorphism iff π is an 1 − 1 mapping. §1.9 Voltage Map 58 Linfan Mao No. 2 Let M = (Xα,β , P) be a map and G a finite group. Call a pair (M, ϑ) a voltage map with group G if ϑ : Xα,β → G, satisfying the following condition: (i) ∀x ∈ Xα,β , ϑ(αx) = ϑ(x), ϑ(αβx) = ϑ(βx) = ϑ−1 (x); (ii) ∀F = (x, y, · · · , z)(βz, · · · , βy, βx) ∈ F (M ), the face set of M , ϑ(F ) = ϑ(x)ϑ(y) · · · ϑ(z) and < ϑ(F )|F ∈ F (u), u ∈ V (M ) >= G, where, F (u) denotes all the faces incident with the vertex u. §1.10 Lifting of a Voltage Map ϑ For a voltage map (M, ϑ) with group G, the map M ϑ = (Xα,β , P ϑ ) is called its lifting map. Theorem 1.3. An finite group G is a fixed-free automorphism group of a map M = (Xα,β , P) on V (M ) iff there is a voltage map (M/G, G) with an assignment ϑ : Xα,β /G → G such that M ∼ = (M/G)ϑ . ( A permutation group G action on Ω is called fixed-free if Gx = 1G for ∀x ∈ Ω.) §1.11 Semi-Arcs of a Graph An edge e = uv ∈ E(Γ) can be divided into two semi-arcs eu , ev . X 12 (Γ)— the set of semi-arcs. Incidence of Semi-Arcs: Call u the root vertex in the semi-arc eu . Two semi-arc eu , fv are said v-incident or e-incident if u = v or e = f . §1.12 A Semi-Arc Automorphism An 1 − 1 mapping ξ on X 12 (Γ) such that ∀eu , fv ∈ X 12 (Γ), ξ(eu ) and ξ(fv ) are v-incident or e-incident if eu and fv are v-incident or e-incident, is called a semi-arc automorphism of the graph Γ. Aut 12 Γ— the semi-arc automorphism group of Γ 1 For ∀g ∈ AutΓ, there is also an induced action g| 2 on X 12 (Γ), g : X 21 (Γ) → X 12 (Γ), as follows: ∀eu ∈ X 12 (Γ), g(eu ) = (g(e)g(u) . 1 All induced action of the elements in AutΓ on X 12 (Γ) is denoted by AutΓ| 2 . Notice that AutΓ ∼ = AutΓ| 2 . 1 Theorem 1.4. For a graph Γ without loops, 1 Aut 12 Γ = AutΓ| 2 . Theorem 1.5. For two maps M1 = (Xα,β , P1 ) and M2 = (Xα,β , P2 ) underlying a graph Γ, then (i) M1 , M2 are equivalent iff M1 , M2 are in one orbits of Aut 12 Γ action on X 12 (Γ); (ii)M1 , M2 are isomorphic iff M1 , M2 are in one orbits of Aut 12 Γ× < α > action on Xα,β . Vol. 1 On automorphisms groups of maps, surfaces and Smarandache geometries 59 Part II Automorphisms of Maps and Klein Surfaces §2.1 Relation of Maps with Klein Surfaces Angles incident with a Quadricell: For a map M = (Xα,β , P), x ∈ Xα,β , the permutation pair {(x, Px), (αx, P −1 αx)} is called an angle incident with x. Theorem 2.1. Any automorphism of a map M = (Xα,β , P) is conformal. Theorem 2.2. If M is a locally orientable map of genus q, then AutM is isomorphic to a group of comformal transformations of a compact Klein surface of genus q. (For Riemann surfaces, the same result gotten by Jones and Singerman in 1978.) §2.2 The Euler Characteristic of Lifting Map Theorem 2.3. The Euler characteristic χ(M ϑ ) of the lifting map M ϑ of the voltage map (M, G) is X 1 χ(M ϑ ) = |G|(χ(M ) + (−1 + )), m m∈O(F (M )) where O(F (M )) denotes the order o(F ) set of the faces in M . §2.3 A Group Being That of a Map Theorem 2.4 If a group G, G ¹ AutM , is fixed-free on V (M ), then X |G|(χ(M/G) + (−1 + m∈O(F (M/G)) 1 )) = χ(M ). m Corollary 2.1. If M is an orientable map of genus p, G ¹ AutM is fixed-free on V (M ) and the quotient map M/G with genus γ, then |G| = 2p − 2 P 2γ − 2 + (1 − m∈O(F (M/G)) . 1 m )) Particularly, if M/G is planar, then |G| = 2p − 2 P −2 + (1 − m∈O(F (M/G)) . 1 m )) Corollary 2.2. If M is a non-orientable map of genus q, G ¹ AutM is fixed-free on V (M ) and the quotient map M/G with genus δ, then |G| = q−2 P δ−2+ (1 − m∈O(F (M/G)) . 1 m )) Particularly, if M/G is projective planar, then |G| = −1 + q−2 P m∈O(F (M/G)) (1 − . 1 m )) 60 Linfan Mao No. 2 Theorem 2.5. If a group G, G ¹ AutM , then X χ(M ) + (|Φv (g)| + |Φf (g)|) = |G|χ(M/G), g∈G,g6=1G where, Φv (g) = {v|v ∈ V (M ), v g = v} and Φf (g) = {f |f ∈ F (M ), f g = f }, and if G is fixed-free on V (M ), then X χ(M ) + |Φf (g)| = |G|χ(M/G). g∈G,g6=1G Corollary 2.3. If a finite group G, G ¹ AutM is fixed-free on V (M ) and transitive on F (M ), for example, M is regular and G = AutM , then M/G is an one face map and χ(M ) = |G|(χ(M/G) − 1) + φ(M ) Corollary 2.4. For an one face map M , if G, G ¹ AutM is fixed-free on V (M ), then χ(M ) − 1 = |G|(χ(M/G) − 1), and |G|, especially, |AutM | is an integer factor of χ(M ) − 1. Remark 2.1. For an one face planar map, i.e., the plane tree, the only fixed-free automorphism group on its vertices is the trivial group by the Corollary 2.4. §2.4 The Non-Euclid Area of a Map For a given voltage map (M, G), its non-Euclid area µ(M, G) is µ(M, G) = 2π(−χ(M ) + X m∈O(F (M )) (−1 + 1 )). m Particularly, since any map M can be viewed as a voltage map (M, 1G ), we get the nonEuclid area of a map M µ(M ) = µ(M, 1G ) = −2πχ(M ). Theorem 2.6. ( Riemann-Hurwitz formula) If G ¹ AutM is fixed-free on V (M ), then |G| = µ(M ) . µ(M/G, ϑ) Theorem 2.7. The non-Euclid area µ(∆) of a triangle ∆ on a surface S with internal angles η, θ, σ is µ(∆) = η + θ + σ − π. §2.5 A Combinatorial Refinement of Huriwtz Theorem Graphical property P : Define a graphical property P to be a kind of subgraphs of a graph Γ, such as, regular subgraphs, circuits, trees, stars, wheels, · · · . Call a subset A of Xα,β of M = (Xα,β , P) has the graphical property P if the underlying graph of A has property P . Vol. 1 On automorphisms groups of maps, surfaces and Smarandache geometries 61 A(P, M )— the set of all the A subset with property P in the map M . Theorem 2.8. Let M = (Xα,β , P) be a map. Then for ∀G ¹ AutM , [|v G ||v ∈ V (M )] | |G| |G| | |A||A(P, M )|, where [a, b, · · · ] denotes least common multiple of a, b, · · · . Corollary 2.5. Let T r2 be the set of tours with each edge appearing 2 times. Then for ∀G ¹ AutM , |T | ≥ 1, T ∈ T r2 , ). 2 Let T r1 be the set of tours without repeat edges. Then |G| | (l|T r2 |, l = |T | = |T | ≥ 1, T ∈ T r1 , ). 2 Particularly, denote by φ(i, j) the number of faces in M with facial length i and singular edges j, then |G| | (2l|T r1 |, l = |T | = |G| | ((2i − j)φ(i, j), i, j ≥ 1), where, (a, b, · · · ) denotes the greatest common divisor of a, b, · · · . Corollary 2.6. Let T be the set of trees in the map M . Then for ∀G ¹ AutM , |G| | (2ltl , l ≥ 1), where tl denotes the number of trees with l edges. Corollary 2.7. Let vi be the number of vertices with valence i. Then for ∀G ¹ AutM , |G| | (2ivi , i ≥ 1). Theorem 2.9. Let M be an orientable map of genus g(M ) ≥ 2. Then for ∀G ¹ Aut+ M , |G| ≤ 84(g(M ) − 1) and for ∀G ¹ AutM , |G| ≤ 168(g(M ) − 1). Corollary 2.8. For any Riemann surface S of genus g ≥ 2, 4g(S) + 2 ≤ |Aut+ S| ≤ 84(g(S) − 1) 8g(S) + 4 ≤ |AutS| ≤ 168(g(S) − 1). Theorem 2.10. Let M be a non-orientable map of genus g 0 (M ) ≥ 3. Then for ∀G ¹ Aut+ M , |G| ≤ 42(g 0 (M ) − 2) 62 Linfan Mao No. 2 and for ∀G ¹ AutM , |G| ≤ 84(g 0 (M ) − 2), with the equality hold iff M is a regular map with vertex valence 3 and face valence 7 or vice via. Corollary 2.9. For any Klein surface K underlying a non-orientable surface of genus q ≥ 3, |Aut+ K| ≤ 42(q − 2) and |AutK| ≤ 84(q − 2). §2.6 The Cyclic Group of a Klein Surface Theorem 2.11. Let M = (Xα,β , P) be a map and N = pr11 · · · prkk , p1 < p2 < · · · < pk , be the arithmetic decomposition of the integer N . Then for any voltage assignment ϑ : Xα,β → ZN , (i) if M is orientable, the minimum genus gmin of the lifting map M ϑ which admits an automorphism of order N , fixed-free on V (M ϑ ), is X gmin = 1 + N {g(M ) − 1 + (1 − m∈O(F (M )) 1 φ(M ) )b c}. p1 2 0 (ii) if M is non-orientable, the minimum genus gmin of the lifting map M ϑ which admits ϑ an automorphism of order N , fixed-free on V (M ), is 0 gmin = 2 + N {g(M ) − 2 + 2(1 − 1 φ(M ) )b c}. p1 2 \ Theorem 2.12. The maximum order Nmax of an automorphism g of an orientable map M of genus≥ 2 is Nmax ≤ 2g(M ) + 1 0 of anautomorphism g of a non-orientable map of genus≥ 3 is and the maximum order Nmax 0 Nmax ≤ g(M ) + 1, where g(M ) is the genus of the map M . Corollary 2.10. The maximum order of an automorphism of a Riemann surface of genus≥ 2 is 2g(M ) + 1, and the maximum order of an automorphism of a non-orientable Klein surface of genus≥ 3 without boundary is g(M ) + 1. §2.7 The Subgroup of a Graph Being That of Maps Theorem 2.13. Let Γ be a connected graph. If G ¹ AutΓ, then G is an automorphism group of a map underlying the graph Γ iff for ∀v ∈ V (Γ), the stabler Gv ¹ < v > × < α >. Theorem 2.14. Let Γ be a connected graph. If G ¹ AutΓ, then G is an orientationpreserving automorphism group of a map underlying the graph Γ iff for ∀v ∈ V (Γ), the stabler Gv ¹ < v > is a cyclic group. Theorem 2.15. Let M be a map underlying the graph G and omax (M, g), omax (G, g) be the maximum order of orientation-preserving automorphism in AutM and in Aut 21 G. Then omax (M, g) ≤ omax (G, g), Vol. 1 On automorphisms groups of maps, surfaces and Smarandache geometries 63 and the equality hold for at least one map underlying the graph G. Corollary 2.11. For any positive integer n, there exists a vertex transitive map M underlying a circultant such that Zn is an orientation-preserving automorphism group of the map M. Corollary 2.12. The maximum order of an orientation - preserving automorphism of a complete map Kn , n ≥ 3, is at most n. Part III The representation of Automorphisms of a Map §3.1 Complete Maps A map underlying a complete graph Kn is called a complete map. Let Kn be a complete graph of order n. Label its vertices by integers 1, 2, ..., n. Then its edge set is {ij|1 ≤ i, j ≤ n, i 6= j ij = ji}, and Xα,β (Kn ) = S {ij+ : 1 ≤ i, j ≤ n, i 6= j} {ij− : 1 ≤ i, j ≤ n, i 6= j} Y α= (ij+ , ij− ), 1≤i,j≤n,i6=j β= Y (ij+ , ij+ )(ij− , ij− ). 1≤i,j≤n,i6=j Theorem 3.1. All orientation-preserving automorphisms of non-orientable complete maps of order≥ 4 are extended actions of elements in E[s ns ] , E [1,s n−1 s ] , and all orientation-reversing automorphisms of non-orientable complete maps of order≥ 4 are extended actions of elements in n , αE[(2s) 2s ] αE 4 [(2s) 2s ] , αE[1,1,2] , where, Eθ denotes the conjugatcy class containing element θ in the symmetry group Sn . Theorem 3.2. All orientation-preserving automorphisms of orientable complete maps of order≥ 4 are extended actions of elements in E[s ns ] , E [1,s n−1 s ] and all orientation-reversing automorphisms of orientable complete maps of order≥ 4 are extended actions of elements in n , αE[(2s) 2s ] αE 4 [(2s) 2s ] , αE[1,1,2] , 64 Linfan Mao No. 2 where,Eθ denotes the conjugatcy class containing θ in Sn . §3.2 Semi-Regular Maps A graph is semi-regular if it is simple and its automorphism group action on its ordered pair of adjacent vertices is fixed-free and a map is semi-regular if it underlying a semi-regular graph. Theorem 3.3. Let Γ be a semi-regular graph. Then all the automorphisms of orientable maps underlying the graph Γ are g|Xα,β and αh|Xα,β , g, h ∈ AutΓ with h ≡ 0(mod2). and all the automorphisms of non-orientable maps underlying the graph Γ are also g|Xα,β and αh|Xα,β , g, h ∈ AutΓ with h ≡ 0(mod2). §3.3 One Face Maps Theorem 3.4. Let Bn be a bouquet with n edges 1, 2, · · · , n. Then the automorphisms (g; h1 , h2 , · · · , hn ) of orientable maps underlying a Bn , n ≥ 1, are respective (O1) g ∈ E[k nk ] , hi = 1, i = 1, 2, · · · , n; (O2) g ∈ E[k nk ] and if n/k g= Y (i1 , i2 , · · · ik ), i=1 where ij ∈ {1, 2, · · · , n}, n/k ≡ 0(mod2), then hi1 = (1, αβ), i = 1, 2, · · · , nk and hij = 1 f or j ≥ 2; (O3) g ∈ E 2s and if n−2ks [k ,(2k) ] 2k g= 2s Y (n−2ks)/2k Y (i1 , i2 , · · · ik ) i=1 (ej1 , ej2 , · · · , ej2k ), j=1 where ij , ejt ∈ {1, 2, · · · , n}, then hi1 = (1, αβ), i = 1, 2, · · · , s, hil = 1 for l ≥ 2 and hjt = 1 for t = 1, 2, · · · , 2k and the automorphisms (g; h1 , h2 , · · · , hn ) of non-orientable maps underlying a Bn , n ≥ 1, are respective (N 1) g ∈ E[k nk ] , hi = 1, i = 1, 2, · · · , n; (N 2) g ∈ E[k nk ] and if n/k g= Y (i1 , i2 , · · · ik ), i=1 where ij ∈ {1, 2, · · · , n}, n/k ≡ 0(mod2), then hi1 = (1, αβ), (1, β) with at least one hi0 1 (1, β), i = 1, 2, · · · , nk and hij = 1 f or j ≥ 2; (N 3) g ∈ E 2s and if n−2ks [k ,(2k) 2k g= ] 2s Y i=1 (n−2ks)/2k (i1 , i2 , · · · ik ) Y j=1 (ej1 , ej2 , · · · , ej2k ), Vol. 1 65 On automorphisms groups of maps, surfaces and Smarandache geometries where ij , ejt ∈ {1, 2, · · · , n}, then hi1 = (1, αβ), (1, β) with at least one hi0 1 = (1, β), i = 1, 2, · · · , s, hil = 1 for l ≥ 2 and hjt = 1 f or t = 1, 2, · · · , 2k. Part IV The Enumeration of Unrooted Maps §4.1 A Scheme for Enumeration Theorem 4.1. For a given graph Γ, let E ⊂ E L (Γ), then the numbers n(E, Γ) and η(E, Γ) of non-isomorphic unrooted maps and non-equivalent embeddings in E are respective 1 2|Aut 12 Γ| n(E, Γ) = X |Φ1 (g)|, g∈Aut 1 Γ 2 where, Φ1 (g) = {P|P ∈ E and P g = P or P gα = P} and η(E, Γ) = 1 |Aut 12 Γ| X |Φ2 (g)|, g∈Aut 1 Γ 2 where, Φ2 (g) = {P|P ∈ E and P g = P}. Corollary 4.1. The numbers nO (Γ), nN (Γ) and nL (Γ) of non-isomorphic unrooted orientable maps ,non-orientable maps and locally orientable maps underlying a graph Γ are respective X 1 nO (Γ) = |ΦO 1 (g)|; 2|Aut 12 Γ| g∈Aut 1 Γ 2 nN (Γ) = 1 2|Aut 12 Γ| X |ΦN 1 (g)|; g∈Aut 1 Γ 2 1 nL (Γ) = 2|Aut 12 Γ| X |ΦL 1 (g)|, g∈Aut 1 Γ 2 ΦO 1 (g) O g gα N g where, = {P|P ∈ E (Γ) and P = P or P = P}, ΦN 1 (g) = {P|P ∈ E (Γ) and P = P L g gα or P gα = P}, ΦL = P}. 1 (g) = {P|P ∈ E (Γ) and P = P or P §4.2 The Number of Complete Maps Theorem 4.2. The number nL (Kn ) of complete maps of order n ≥ 5 on surfaces is nL (Kn ) = 1 X ( + 2 k|n X n ) k|n,k≡0(mod2) 2α(n,k) (n − 2)! k + n k k ( nk )! X k|(n−1),k6=1 where,   α(n, k) = and  n(n−3) 2k , n(n−2) 2k , if k ≡ 1(mod2); if k ≡ 0(mod2), φ(k)2β(n,k) (n − 2)! n−1 n−1 k , 66 Linfan Mao   β(n, k) =  (n−1)(n−2) , 2k (n−1)(n−3) , 2k No. 2 if k ≡ 1(mod2); if k ≡ 0(mod2). and nL (K4 ) = 11. Theorem 4.3. The number nO ((Kn ) of complete maps of order n ≥ 5 on orientable surfaces is nO (Kn ) = 1 X ( + 2 k|n X k|n,k≡0(mod2) n ) (n − 2)! k + n k k ( nk )! and n(K4 ) = 3. For K4 on the surfaces, see the Fig.2 Fig.2 ¸ §4.3 The Number of Semi-Regular Maps X k|(n−1),k6=1 φ(k)(n − 2)! n−1 n−1 k . Vol. 1 67 On automorphisms groups of maps, surfaces and Smarandache geometries Theorem 4.4. Let Γ be a semi-regular graph. Then the numbers of unrooted maps on orientable and non-orientable surfaces underlying the graph Γ are nO (Γ) = X Y d(x) 1 ( λ(ξ) ( − 1)! |AutΓ| o(ξ| NΓ (x) ) V ξ∈AutΓ x∈Tξ and nN (Γ) = 1 |AutΓ| X × E (2|Tξ |−|TξV | − 1)λ(ξ) ξ∈AutΓ Y x∈TξV ( d(x) − 1)!, o(ξ|NΓ (x) ) where λ(ξ) = 1 if o(ξ) ≡ 0(mod2) and 21 , otherwise. Corollary 4.2. Let Γ = Cay(Zp : S) be connected graph of prime order p with (p−1, |S|) = 2. Then O n (Γ, M) = (|S| − 1)!p + 2p(|S| − 1)! 4p p+1 2 + (p − 1)(|S| − 1)! 4p and N n (Γ, M) = + (2 p|S| 2 −p (2 |S|−2 2 − 1)(|S| − 1)!p 2(2 + 2p p|S|−2p−2) 4 − 1)p(|S| − 1)! 2p p+1 2 − 1)(p − 1)(|S| − 1)! . 4p §4.4 The Number of One Vertex Maps Theorem 4.5. The number nO (Bn ) of non -isomorphic maps on orientable surfaces underlying a graph Bn is O n (Bn ) = X k 2n k −1 k|2n,k6=2n + φ(2n) 2n 1 ∂ ( − 1)! 2n k ( k )! 2n k (Z(Sn [S2 ])) 2n ∂skk |sk =0 ∂(Z(Sn [S2 ])) |s2n =0 ∂s2n Theorem 4.6. he number nN (Bn ) of non -isomorphic maps on the non-orientable surfaces with an underlying graph Bn is nN (Bn ) = + (2n − 1)! + n! 1 2n n! X ( s≥1 X k|2n,3≤k<2n (2k) 2n k −1 ( 2n ∂ − 1)! k 2n k (Z(Sn [S2 ])) 2n ∂skk |sk =0 n n! ∂ n (Z(Sn [S2 ])) |s2 =0 − b c)). + 4n (n − 1)!( n (n − 2s)!s! ∂s2 2 For B2 on the surfaces, see the Fig.3. 68 Linfan Mao No. 2 Fig.3 ¸ Part V Map Geometry §5.1 What are the Contribution of Maps to Mathematics Klein Erlanger Program: Any geometry is finding invariant properties under the transformation group of this geometry (This is essentially the group action idea.) The following problems are applications of the Klein Erlanger Program in maps: (i)determine isomorphism maps or rooted maps; (ii)determine equivalent embeddings of a graph; (iii)determine an embedding whether exists; (iv)enumerate maps or rooted maps on a surface; (v)enumerate embeddings of a graph on a surface; (vi) · · · , etc. What are their importance to classical mathematics? What are their contributions to science? §5.2 Smarandache Geometries Classical geometries: The axiom system of Euclid geometry is the following: (A1)there is a straight line between any two points. (A2)a finite straight line can produce a infinite straight line continuously. Vol. 1 On automorphisms groups of maps, surfaces and Smarandache geometries 69 (A3)any point and a distance can describe a circle. (A4)all right angles are equal to one another. (A5)if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. The axiom (A5) can be also replaced by: (A5’)given a line and a point exterior this line, there is one line parallel to this line. The Lobachevshy-Bolyai-Gauss geometry, also called hyperbolic geometry, is a geometry with axioms (A1) − (A4) and the following axiom (L5): (L5) there are infinitely many line parallels to a given line passing through an exterior point. The Riemann geometry is a geometry with axioms (A1) − (A4) and the following axiom (R5): there is no parallel to a given line passing through an exterior point. Smarandache introduced the paradoxist geometry, non-geometry, counter-proje -ctive geometry and anti-geometry by contradicts the axioms (A1) − (A5) in Euclid geometry, generalize the classical geometries. For example, the axioms of a Paradoxist geometry are (A1) − (A4) and with one of the following as the axiom (P 5): (i)there are at least a straight line and a point exterior to it in this space for which any line that passes through the point intersect the initial line. (ii)there are at least a straight line and a point exterior to it in this space for which only one line passes through the point and does not intersect the initial line. (iii)there are at least a straight line and a point exterior to it in this space for which only a finite number of lines l1 , l2 , · · · , lk , k ≥ 2 pass through the point and do not intersect the initial line. (iv)there are at least a straight line and a point exterior to it in this space for which an infinite number of lines pass through the point (but not all of them) and do not intersect the initial line. (v)there are at least a straight line and a point exterior to it in this space for which any line that passes through the point and does not intersect the initial line. F. Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000. The Smarandache geometries are defined as follows. Definition 5.1. An axiom is said Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways. A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969). A nice model for the Smarandache geometries, called s-manifolds, is found by Isier, which is defined by Mao using maps as follows: An s-manifold is any collection C(T, n) of these equilateral triangular disks Ti , 1 ≤ i ≤ n satisfying the following conditions: 70 Linfan Mao No. 2 (i) Each edge e is the identification of at most two edges ei , ej in two distinct triangular disks Ti , Tj , 1 ≤ i, j ≤ n and i 6= j; (ii) Each vertex v is the identification of one vertex in each of five, six or seven distinct triangular disks. H.Iseri, Smarandache manifolds, American Research Press, Rehoboth, NM,2002. L.F.Mao, Automorphism groups of maps, surfaces and Smarandache geometries, American Research Press, Rehoboth, NM,2005. §5.3 A Classification of Smarandache Manifolds Classical Type: (1) ∆1 = {5 − regular triangular maps} (elliptic); (2) ∆2 = {6 − regular triangular maps}(euclidean); (3) ∆3 = {7 − regular triangular maps}(hyperbolic). Smarandache Type: (4) ∆4 = {triangular maps with vertex valency 5 and 6} (euclid-elliptic); (5) ∆5 = {triangular maps with vertex valency 5 and 7} (elliptic-hyperbolic); (6) ∆6 = {triangular maps with vertex valency 6 and 7} (euclid-hyperbolic); (7) ∆7 = {triangular maps with vertex valency 5, 6 and 7} (mixed). Theorem 5.1. |∆1 | = 2, |∆5 | ≥ 2 and |∆i |, i = 2, 3, 4, 6, 7 are infinite. Iseri proposed a question: Do the other closed 2-manifolds correspond to s-manifolds with only hyperbolic vertices?. Since |∆3 | is infinite, the answer is affirmative for this question. §5.4 Map Geometry Definition 5.2. For a combinatorial map M with each vertex valency≥ 3, associates a real number µ(u), 0 < µ(u) < π, to each vertex u, u ∈ V (M ). Call (M, µ) the fundamental map space, µ(u) the angle factor of the vertex u and to be orientable or non-orientable if the map M is orientable or not. Definition 5.3. A point u in a map space (M, µ) is called elliptic, euclidean or hyperbolic if ρ(u)µ(u) < 2π, ρ(u)µ(u) = 2π or ρ(u)µ(u) > 2π. Definition 5.4. Let (M, µ) be a map space. An m-line in (M, µ) is a curve with a constant curvature. Points in (M, µ) are called m-points. We have the following result for map geometries. Theorem 5.2. For any planar map M with order≥ 3 and vertex valency≥ 3, there is an angle factor µ such that (M, µ) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5), (L5) and (R5). Vol. 1 On automorphisms groups of maps, surfaces and Smarandache geometries 71 Fig.4 ¸ Theorem 5.3. For any map M on an orientable surface with order≥ 3 and vertex valency≥ 3, there is an angle factor µ such that (M, µ) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5),(L5) and (R5). Theorem 5.4. Let P be a k-polygon in a map space with each line segment passes through at most one elliptic or hyperbolic point. If H is the set of elliptic points and hyperbolic points on the line segment of P , then the sum of the internal angles in P is 1 X (k + |H| − 2)π − ρ(u)µ(u). 2 u∈H Corollary 5.1. Let 4 be a triangle in a map space. Then (i) if 4 is euclidean, then then the sum of its internal angles is equal to π; (ii) if 4 is elliptic, then the sum of its internal angles is less than π; (iii) if 4 is hyperbolic, then the sum of its internal angles is more than π. Theorem 5.5. The number nO (Γ, g) of non -equivalent orientable map geometries underlying a simple graph Γ by denial the axiom (A5) by (A5), (L5) or (R5) is Q 3|Γ| (ρ(v) − 1)! v∈V (Γ) 2|AutΓ| where ρ(v) is the valency of the vertex v in the graph G. , 72 Linfan Mao No. 2 Part VI Open Problems for Combinatorial Maps §6.1 The Uniformization Theorem for Simple Connected Riemann Surfaces The uniformization theorem for simple connected Riemann surfaces is one of those beautiful results in the Riemann surface theory, which is stated as follows: If S is a simple connected Riemann surface, then S is conformally equivalent to one and only one of the following three: S (a) C ∞; (b) C; (c) 4 = {z ∈ C||z| < 1}. How can we define the conformal equivalence for maps enabling us to get the uniformization theorem of maps? What is the correspondence class maps with the three type (a) − (c) Riemann surfaces? §6.2 Combinatorial Construction of an Algebraic Curve of Genus A complex plane algebraic curve Cl is a homogeneous equation f (x, y, z) = 0 in P2 C = 2 (C \ (0, 0, 0))/ ∼, where f (x, y, z) is a polynomial in x, y and z with coefficients in C. The degree of f (x, y, z) is said the degree of the curve Cl . For a Riemann surface S, a well-known result is ([2]) there is a holomorphic mapping ϕ : S → P2 C such that ϕ(S) is a complex plane algebraic curve and g(S) = (d(ϕ(S)) − 1)(d(ϕ(S)) − 2) . 2 By map theory, we know a combinatorial map also is on a surface with genus. Then whether we can get an algebraic curve by all edges in a map or by make operations on the vertices or edges of the map to get plane algebraic curve with given k-multiple points? how do we find the equation f (x, y, z) = 0? §6.3 Classification of s-Manifolds by Maps We present an elementary classification for the closed s-manifolds in the Part V . For the general s-manifolds, their correspondence combinatorial model is the maps on surfaces with boundary, founded by Bryant and Singerman in 1985 (R.P.Bryant and D.Singerman, Foundations of the theory of maps on surfaces with boundary,Quart.J.Math.Oxford(2),36(1985), 17-41.). The later is also related to the modular groups of spaces and need to investigate further itself. The questions are (i) how can we combinatorially classify the general s-manifolds by maps with boundary? (ii) how can we find the automorphism group of an s-manifold? (iii) how can we know the numbers of non-isomorphic s-manifolds, with or without root? Vol. 1 On automorphisms groups of maps, surfaces and Smarandache geometries 73 (iv) find rulers for drawing an s-manifold on a surface, such as, the torus, the projective plane or Klein bottle, not the plane. §6.4 Map Geometries (i) For a given graph, determine properties of the map geometries underlying this graph. (ii) For a given locally orientable surface, determine the properties of map geometries on this surface. (iii) Classify map geometries on a locally orientable surface. (iv) Enumerate non-equivalent map geometries underlying a graph or on a locally orientable surface. (v) Establish the surface geometry by map geometries. (vi) Applying map geometries to classical mathematics or other sciences. §6.5 Gauss Mapping Among Surfaces In the classical differential geometry, a Gauss mapping among surfaces is defined as follows: Let S ⊂ R3 be a surface with an orientation N. The mapping N : S → R3 takes its value in the unit sphere S 2 = {(x, y, z) ∈ R3 |x2 + y 2 + z 2 = 1} along the orientation N. The map N : S → S 2 , thus defined, is called the Gauss mapping. we know that for a point P ∈ S such that the Gaussian curvature K(P ) 6= 0 and V a connected neighborhood of P with K does not change sign, N (A) , A→0 A where A is the area of a region B ⊂ V and N (A) is the area of the image of B by the Gauss mapping N : S → S 2 . The questions are (i) what is its combinatorial meaning of the Gauss mapping? How to realizes it by maps? (ii) how we can define various curvatures for maps and rebuilt the results in the classical differential geometry? K(P ) = lim §6.6 The Gauss-Bonnet Theorem Let S be a compact orientable surface. Then Z Z Kdσ = 2πχ(S), S where K is Gaussian curvature on S. This is the famous Gauss-Bonnet theorem for compact surface. The questions are (i) what is its combinatorial mean of the Gauss curvature? (ii) how can we define the angle, area, volume, curvature, · · · , of a map? (iii)can we rebuilt the Gauss-Bonnet theorem by maps? or can we get a generalization of the classical Gauss-Bonnet theorem by maps? Scientia Magna Vol. 1 (2005), No. 2, 74-77 On the mean value of Smarandache ceil 1 function Ding Liping Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China, 710069 Abstract For any fixed positive integer n, the Smarandache ceil function of order k is denoted by N ∗ → N and has the following definition: Sk (n) = min{x ∈ N | n | xk } (∀n ∈ N ∗ ) . In this paper, we study the mean value properties of the Smarandache ceil function, and give a sharp asymptotic formula for it. Keywords Smarandache ceil function; Mean value; Asymptotic formula. §1. Introduction For any fixed positive integer n, the Smarandache ceil function of order k is denoted by N → N and has the following definition: ∗ Sk (n) = min{x ∈ N | n | xk } (∀n ∈ N ∗ ) . For example, S2 (1) = 1, S2 (2) = 2, S2 (3) = 3, S2 (4) = 2, S2 (5) = 5, S2 (6) = 6, S2 (7) = 7, S2 (8) = 4, S2 (9) = 3, · · · . This was introduced by Smarandache who proposed many problems in [1]. There are many papers on the Smarandache ceil function. For example, Ibstedt [2] and [3] studied this function both theoretically and computationally, and got the following conclusions: (∀a, b ∈ N ∗ )(a, b) = 1 ⇒ Sk (ab) = Sk (a)Sk (b), α1 αr αr 1 α2 Sk (pα 1 p2 . · · · .pr ) = S( p1 ). · · · .S( pr ). In this paper, we study the mean value properties of the Smarandache ceil function, and give a sharp asymptotic formula for it. That is, we shall prove the following: Theorem. Let x ≥ 2, for any fixed positive integer k, we have the asymptotic formula ¶¸ µ ³ 3 ´ X Y· x2 1 1 + O x 2 +² . Sk (n) = ζ(2k − 1) 1− 1 + 2k−3 2 p(p + 1) p p n≤x where ζ(s) is the Riemann zeta function, Y denotes the product over all prime p, and ² be p any fixed positive number. 1 This work is supported by the N.S.F(60472068) and P.N.S.F of P.R.China Vol. 1 75 On the mean value of Smarandache ceil function This solved a conjecture of [4]. From this theorem we may immediately deduce the following: Corollary 1. For any real number x ≥ 2, we have the asymptotic formula: X S2 (n) = n≤x ³ 3 ´ 3x2 ζ(3) + O x 2 +² . π2 Corollary 2. Let x ≥ 1 and Sc(n) denotes the smallest cube greater than or equal to n, then we have ³ 4´ X 9 5 (Sc(n) − n) = x3 + O x3 . 10 n≤x §2. A Lemma To complete the proof of the theorem, we need the following famous Perron formula [5]: ∞ X Lemma. Suppose that the Dirichlet series f (s) = a(n)n−s , with s = σ + it is convern=1 gent absolutely for σ > β, and that there exist a positive λ and a positive increasing function A(s) such that ∞ X |a(n)| n−σ ¿ (σ − β)−1 , σ → β + 0 n=1 and a(n) ¿ A(n), n = 1, 2, · · · . Then for any b > 0, b + σ > β, and x not to be an integer, we have X a(n)n n≤x −s0 Z b+iT xω = f (s0 + ω) dω + O ω b−iT ¶ µ A(2x)x1−σ log x , +O T || x || 1 2πi µ where || x || is the nearest integer to x. §3. Proof of the theorem In this section, we shall complete the proof of Theorem. Let f (s) = where Re(s) > 3. ∞ X Sk (n) , ns n=1 xb T (b + σ − β)λ ¶ 76 Ding Liping No. 2 By Euler product formula [6], we have ¶ Yµ Sk (p) Sk (p2 ) Sk (pk ) f (s) = 1+ + + ··· + + ··· ps p2s pks p ¶ Yµ p p p2 p2 p = 1 + s + 2s + · · · + ks + (k+1)s + · · · + 2ks + · · · p p p p p p à ! 1 1 Y 1 1 − pks p2 1 − pks = 1 + s−1 + (k+1)s + ··· 1 p p 1 − ps 1 − p1s p ! à 1 Y 1 − p1ks ps−1 = 1+ 1 1 − p1s 1 − pks−1 p à µ ¶! ζ(s)ζ(s − 1)ζ(ks − 1) Y 1 1 1 = 1− + s 1 ζ(2s − 2) pks−1 p 1 + ps−1 p where ζ(s) is the Riemann zeta function. 5 Taking s0 = 0, b = 3, T = x 2 in the Lemma, we have Z 3+iT X 3 xs 1 ζ(s)ζ(s − 1)ζ(ks − 1) R(s) ds + O(x 2 +ε ), Sk (n) = 2iπ 3−iT ζ(2s − 2) s n≤x where R(s) = Y à 1− p To estimate the main term 1 2iπ Z 3+iT µ 1 1+ 1 ps−1 pks−1 1 + s p ¶! . ζ(s)ζ(s − 1)ζ(ks − 1) xs R(s) ds, ζ(2s − 2) s 3−iT we move the integral line from s = 3 ± iT to s = f (s) = 1 3 2 ± iT . This time, the function ζ(s)ζ(s − 1)ζ(ks − 1)xs R(s) ζ(2s − 2)s 2 has a simple pole point at s = 2 with residue x2 ζ(2k − 1)R(2). So we have ÃZ Z 32 +iT Z 32 −iT Z 3−iT ! 3+iT 1 ζ(s)ζ(s − 1)ζ(ks − 1)xs + + + R(s)ds 3 3 2iπ ζ(2s − 2)s 3−iT 3+iT 2 +iT 2 −iT µ ¶¸ Y· 1 x2 1 ζ(2k − 1) 1 + 2k−3 . = 1− 2 p(p + 1) p p Note that 1 2iπ ÃZ 3 2 +iT 3+iT Z + 3 2 −iT 3 2 +iT Z 3−iT + 3 2 −iT ! 3 ζ(s)ζ(s − 1)ζ(ks − 1)xs R(s)ds ¿ x 2 +² ζ(2s − 2)s From the above, we may immediately get the asymptotic formula: ¶¸ µ ³ 3 ´ X Y· x2 1 1 + O x 2 +² . Sk (n) = ζ(2k − 1) 1− 1 + 2k−3 2 p(p + 1) p p n≤x This completes the proof of Theorem. Vol. 1 On the mean value of Smarandache ceil function 77 References [1] F.Smarandache, Only problems, Not solutions, Chicago, Xiquan Publ. House, 1993. [2] Ibstedt, Surfing on the Ocean of Numbers-A Few Smarandache Notions and Similar Topics, Erhus University press, New Mexico,USA. 1997 [3] Ibstedt, Computational Aspects of Number Sequences, American Research Press, Lupton USA, 1999 [4] S.Tabirca and T. Tabirca, Smarandache Notions Journal, 13, 2002, 30-36 [5] Pan Chengdong and Pan Chengbiao, Goldbach conjecture, Science Press, Beijing, 1992, 145 [6] Tom M.Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976 Scientia Magna Vol. 1 (2005), No. 2, 78-80 An equation concerning the Smarandache function1 Maohua Le Department of Mathematics, Zhanjiang Normal College 29 Cunjin Road, Chikan Zhanjiang, Guangdong, P. R. China Abstract In this paper, we solve an open question concerning the Smarandache function. Keywords Smarandache function; Euler totient function; Diophantine equation. For any positive integer n, let S(n) and ϕ(n) denote the Smarandache function and the Euler totient function respectively. In [1], Bencze asked that solve the equation à n ! n X Y k (1) S n = ϕ(n) S(k) k=1 k=1 in positive integers n. In this paper we solve this question as follows. Theorem. The equation (1) has only positive integer solution n = 1. The proof of our Theorem depends on the following lemmas. Lemma 1([3]). If a is a positive integer with a > 1, then S(a) > 1. Lemma 2([3]). If a and b are coprime positive integers, then we have S(ab) = max(S(a), S(b)). Lemma 3([2]). If p is a prime and α is a positive integer, then we have S(pα ) ≤ pα and p|S(pα ). Proof of Theorem. It is easy to see that (1) has only solution n = 1 with n ≤ 5. We now ¡ ¢δ suppose that n is a positive integer solution of (1) with n > 5. Since gcd n, 1 + n + · · · + nn−1 = 1, by Lemma 2, we get à n ! X ¡ ¡ ¢¢ S nk = S n 1 + n + · · · + nn−1 (2) k=1 ¡ ¢ = max S(n), S(1 + n + · · · + nn−1 ) . If S(n) ≥ S(1 + n + · · · + nn−1 ), then from (1) and (2) we obtain 1 = ϕ(n) n−1 Y S(k). (3) k=1 Since n ≥ 5, by Lemma 1, we get S(n − 1) > 1 and (3) is impossible. So we have S(n) < S(1 + n + · · · + nn−1 ). This work is supported by N.S.F. of P. R. China(No.10271104), the Guangdong Provincial Natural Science and the Natural Science Foundation of the Education Department of Guangdong Province(No.0161). 1 Foundation(No.011781) Vol. 1 79 An equation concerning the Smarandache function Then, by (1) and (2), we get S(1 + n + · · · + nn−1 ) = ϕ(n) n Y S(k). (4) k=1 Let αr 1 α2 1 + n + · · · + nn−1 = pα 1 p2 · · · pr (5) be the factorization of 1 + n + · · · + nn−1 . By Lemma 2, we have α2 αr 1 S(1 + n + · · · + nn−1 ) = max(S(pα 1 ), S(p2 ), · · · , S(pr )). (6) S(1 + n + · · · + nn−1 ) = S(pα ), (7) It implies that where α p α = pj j , 1 ≤ j ≤ r. (8) Hence, by (1) and (7), we get S(pα ) = ϕ(n) n Y S(n). (9) k=1 Since p is a prime, we find from (9) that p|S(pα ) and p|ϕ(n) or p|S(k), 1 ≤ k ≤ n. (10) On the other hand, by Lemma 3, we have S(pα ) ≤ αp. Therefore, we get from (9) that αp ≥ ϕ(n) n Y S(k). (11) k=1 Since n ≥ 5, we have ϕ(n) > 1 and S(k) > 1 for k = 2, · · · , n. hence, by (10) and (11), we get n α≥ Y 1 ϕ(n) S(k) > 2n−1 . p (12) k=1 However, since 1 + n + · · · + nn−1 is odd, we see from (5) and (8) that pα < nn and α< n log n n log n < < n log n. log p log 3 (13) The combination of (12) and (13) yields n log n > 2n−1 , n > 5, a contradiction. Thus, (1) has only solution n = 1. The theorem is proved. (14) 80 Maohua Le No. 2 References [1] M.Bencze, Open questions for the Smarandache function, Smarandache Notions Journal. 12(2001), 201-203. [2] M.Farris and P.Mitchell, Bounding the Smarandache function, Smarandache Notions Journal. 13(2002), 37-42. [3] F.Smarandache, A function in number theory, An. Univ. Timisoara, 18(1980). Scientia Magna Vol. 1 (2005), No. 2, 81-88 An extension of ABC-theorem1 Morteza Bayat , Hossein Teimoori and Mehdi Hassani Institute for Advanced Studies in Basic Sciences P.O. Box 45195-159 Zanjan, Iran. email: bayat, teimoori, mhassani@iasbs.ac.ir Abstract In this paper we generalize the abc-theorem for n-polynomials over F[x] in which F is an algebraically closed field of characteristic zero. This generalization is obtained by considering the Wronskian of functions over F[x]. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) mn−1 n 1 2 am + am + · · · + an−1 = am n , 1 2 where a1 , a2 , · · · , an ∈ F[x] such that at most one of ai ’s is constant, and m1 , m2 , · · · , mn ∈ N, has no solution for which ai (i = 1, · · · , n) are relatively prime by pairs provided that n(n − 2) ≤ min {mi }. 1≤i≤n Keywords tions. abc-theorem; abc-conjecture; algebraically closed field; Wronskian; Diophantine equa- §1. Introduction Although the arithmetic abc-conjecture is a great mystery, its algebraic counterpart is a rather easy theorem (abc-theorem). It looks like it was first noticed by W.W. Stothers [1]. Later on it was generalized and rediscovered independently by several people, including R.C. Mason [2] and J.H. Silverman [3]. Discovering the abc-theorem, opened a new way for investigating the Fermat’s last theorem over the polynomials with coefficients in an algebraically closed field of characteristic zero. This theorem presented a very elementary proof of the Fermat’s last theorem for polynomials. This led mathematician to give a variant of this theorem over the ring of integer numbers. Of course, this result has been stated as a conjecture and this conjecture has not been proved yet. Today this conjecture is known as the abc-conjecture. Let us state the original abc-theorem [1-4,8,9]. To do this, we need to introduce some notations. We denote the set of all polynomials of one variable x over F by F[x], where F is an algebraically closed field of characteristic zero. We also consider the non-zero elements of F[x], as follows f (x) = c r Y (x − αi )mi , i=1 where α1 , α2 , · · · , αr are the distinct roots of f , c 6= 0 is a constant, and the positive integers mi (i = 1, 2, · · · , r) are the multiplicities of the roots. The degree of the polynomial f is deg f = m1 + m2 + · · · + mr . 1 The first and second authors are supported in part by the Institute for Advanced Studies in Basic Sciences Zanjan, IRAN 82 Morteza Bayat , Hossein Teimoori and Mehdi Hassani No. 2 The number of distinct roots of f will be denoted by n0 (f ). Thus, we have n0 (f ) = r. If f, g are two nonzero polynomials, then in general n0 (f g) ≤ n0 (f ) + n0 (g), and the equality holds whenever f, g are relatively prime. Now, we state the abc-theorem. The abc-Theorem (Stothers, Mason, Silverman). Let a, b, c ∈ F[x] be non-constant relatively prime polynomials satisfying a + b = c. Then max{deg a, deg b, deg c} ≤ n0 (abc) − 1. The similar result for the ring of integers is well-known as the abc-conjecture. This conjecture has been stated by Oesterle and Masser [5,6] in 1986. The abc-Conjecture (Oesterle, Masser). Given ε > 0, there exists a constant C(ε) such that for all a, b, c ∈ Z with a + b = c, we have the inequality max{|a|, |b|, |c|} ≤ C(ε)(N0 (abc))1+ε , in which N0 (abc) denotes the radical of abc. By radical function we mean Y N0 (n) = p (p is prime and n ∈ N). p|n Note that Stewart and Tijdeman gave some lower bounds for C(ε) (cf [7]). §2. Generalizing ABC-Theorem Now, we generalized the abc-theorem for n-functions. To do this, we need the following lemmas: Lemma 1. Suppose f is a nonzero polynomial in F[x]. Then, we have deg f − m.n0 (f ) ≤ deg(f, f 0 , · · · , f (m) ), (1) where (f, f 0 , · · · , f (m) ) is the greatest common divisor of f, f 0 , · · · , f (m) . Needless to say that the derivative is considered as a purely algebraic operator over the elements of F[x]. However, all known rules for derivatives in calculus text book can be easily proved by means of simple algebraic tools. Qr Proof of Lemma 1. Suppose f (x) = c i=1 (x − αi )mi , in which α1 , α2 , · · · , αr are the distinct roots of f with multiplicities m1 , m2 , · · · , mr respectively. Case I. Suppose for any i(1 ≤ i ≤ r) we have mi ≤ m. Then we get deg f = r X mi ≤ mr = m.n0 (f ) ≤ m.n0 (f ) + deg(f, f 0 , · · · , f (m) ). i=1 Case II. Now, we suppose that there exists an i such that mi > m. Therefore, we have ¯ (x − αi )mi −m ¯ f (j) (j = 0, 1, · · · , m), Vol. 1 83 An Extension of ABC-theorem and consequently, ¯ (x − αi )mi −m ¯ (f, f 0 , · · · , f (m) ). It is clear to see that, Y ¯ (x − αi )mi −m ¯ (f, f 0 , · · · , f (m) ). 0<mi −m 1≤i≤r Considering the degrees of the both sides of the above result, we obtain X (mi − m) ≤ deg(f, f 0 , · · · , f (m) ). 0<mi −m 1≤i≤r Since r X i=1 we get r X (mi − m) ≤ X (mi − m), 0<mi −m 1≤i≤r (mi − m) ≤ deg(f, f 0 , · · · , f (m) ), i=1 or equivalently deg f − m.n0 (f ) ≤ deg(f, f 0 , · · · , f (m) ), and this completes our proof. Remark 1. If char(F) = 0, then we conclude that deg f − m.n0 (f ) ≤ deg(f, f (m) ) = X (mi − m). 0<mi −m 1≤i≤r Definition 1. Let f1 , f2 , · · · , fn be functions over the ring F[x]. The Wronskian of these functions is defined by, ¯ ¯ ¯ (i−1) ¯ W [f1 , f2 , · · · , fn ] = det ¯fj . ¯ 1≤i,j≤n Lemma 2. If char(F) = 0 and f1 , f2 , · · · , fn be linearly independent functions over F in F[x], then there exists an element x in F, such that W [f1 , f2 , · · · , fn ](x) 6= 0 ( i.e. W [f1 , f2 , · · · , fn ](x) is a nonzero polynomial ). Proof. Suppose for every x ∈ F, we have W [f1 , f2 , · · · , fn ](x) = 0. Therefore, there are constant numbers ci (i = 1, 2, · · · , n) in F, such that at least one of these ci is nonzero and       0 fn (x) f1 (x)             0 0  fn (x)   0   f1 (x)   =  ,  + · · · + cn  c1  .. ..   ..       .     . .       (n−1) (n−1) 0 fn (x) f1 (x) 84 Morteza Bayat , Hossein Teimoori and Mehdi Hassani No. 2 or c1 f1 (x) + · · · + cn fn (x) = 0, which is a contradiction with the linearly independence of f1 , f2 , · · · , fn . Lemma 3. Suppose char(F) = 0 and f1 , f2 , · · · , fn are nonzero functions in F[x]. Then, for W [f1 , f2 , · · · , fn ] 6= 0, we have n(n − 1) . 2 (2) n(n − 1) − 1. 2 (3) deg W [f1 , f2 , · · · , fn ] ≤ deg(f1 f2 · · · fn ) − Whenever deg f1 = · · · = deg fn , we get deg W [f1 , f2 , · · · , fn ] ≤ deg(f1 f2 · · · fn ) − Proof. We proceed it by mathematical induction on n. The initialization step n = 1, is clear. Suppose it holds for n − 1 nonzero functions. By expanding the Wronskian determinant W [f1 , f2 , · · · , fn ] with respect to the first row, we obtain W [f1 , f2 , · · · , fn ] = n X 0 0 (−1)i+1 fi .W [f10 , · · · , fi−1 , fi+1 , · · · , fn0 ]. (4) i=1 We have the following inequality for degrees 0 0 deg W [f1 , f2 , · · · , fn ] ≤ max {deg fi + deg W [f10 , · · · , fi−1 , fi+1 , · · · , fn0 ]}, 1≤i≤n and since W [f1 , f2 , · · · , fn ] 6= 0, there exists an i such that the right-hand side has the greatest degree, namely 0 0 deg W [f1 , f2 , · · · , fn ] ≤ deg fi + deg W [f10 , · · · , fi−1 , fi+1 , · · · , fn0 ]. (5) Now, considering the induction hypothesis for the set of (n − 1)-functions 0 0 f10 , · · · , fi−1 , fi+1 , · · · , fn0 , we get (n − 1)(n − 2) 2 n(n − 1) ≤ deg(f1 · · · fi−1 fi+1 · · · fn ) − . 2 0 0 0 0 deg W [f10 , · · · , fi−1 , fi+1 , · · · , fn0 ] ≤ deg(f10 · · · fi−1 fi+1 · · · fn0 ) − (6) Finally, by (5) and (6), we have deg W [f1 , f2 , · · · , fn ] ≤ deg(f1 f2 · · · fn ) − n(n − 1) . 2 For proving (3), it is necessary to show that after expanding the determinant of W [f1 , f2 , · · · , fn ], the term with the highest degree is vanished. We prove this by induction on n, with n ≥ 2. First we investigate the case n = 2. Since deg f1 = deg f2 , we have f1 (x) = ak xk + P (x) and Vol. 1 An Extension of ABC-theorem 85 f2 (x) = bk xk + Q(x), where P (x) and Q(x) are two polynomials of degree at most (k − 1). So, we have ¯ ¯ ¯ ¯ ¯ ak xk + P (x) bk xk + Q(x) ¯ ¯ ¯ W [f1 , f2 ] = ¯ ¯ ¯ kak xk−1 + P 0 (x) kbk xk−1 + Q0 (x) ¯ = ak xk Q0 (x) + kbk xk−1 P (x) − bk xk P 0 (x) − kak xk−1 Q(x). Now, assume its validity for any arbitrary (n − 1)-functions. Then the proof is straight forward considering the relation (4). Now, we are ready to state our main result Theorem 1. Let fn = f1 + f2 + · · · + fn−1 , in which fi ’s are relatively prime by pairs in F[x] with char(F) = 0 and at most one of them is constant. Then, we have (n − 1)(n − 2) , 2 (7) (n − 1)(n − 2) − 1. 2 (8) max deg fi ≤ (n − 2)n0 (f1 f2 · · · fn ) − 1≤i≤n and also min deg fi ≤ (n − 2)n0 (f1 f2 · · · fn ) − 1≤i≤n Proof. For proving the first inequality, we distinguish between two cases. The proof of Case I, is analogous with [9, Theorem 1.2]. Case I. Let f1 , f2 , · · · , fn−1 be linearly dependent over F. Now, the proof proceeds by induction on n. For n = 3, it is true; considering the results in [1-4]. Assume that the theorem is true for all cases n0 , 3 ≤ n0 < n, and consider n polynomials. In equality fn = f1 +f2 +· · ·+fn−1 , assume that fi (i = 1, 2, · · · , n − 1), are linearly dependent over F. Note that, at most one of the fi (i = 1, 2, · · · , n − 1), is constant. Let {fi1 , · · · , fiq }, q < n − 1, be a maximal linearly independent subset of the fi (i = 1, 2, · · · , n − 1). Since n − 1 ≥ 2, and fj ’s are relatively prime by pairs, it follows that q ≥ 2. So each fj , 1 ≤ j ≤ n − 1; j not one of the ik , is a linear combination of the fik , of the form (9) fj = λ1 fi1 + · · · + λq fiq , where the λk ∈ F, and at least two of these λk are not zero. Using our inductive hypothesis we apply the theorem to (9). This yields that if λk 6= 0, then q Y deg fik ≤ (q − 1)n0 (fj fik ) − k=1 q(q − 1) , 2 and so that deg fik ≤ (q − 1)n0 ( n Y i=1 fi ) − q(q − 1) . 2 (10) Qn Now, since at most one of fi is a constant, i.e. n − 1 ≤ n0 ( i=1 fi ), we yield that (q − 1)n0 ( n Y i=1 fi ) − n Y q(q − 1) (n − 1)(n − 2) ≤ (n − 2)n0 ( fi ) − . 2 2 i=1 (11) 86 Morteza Bayat , Hossein Teimoori and Mehdi Hassani No. 2 Now, using (10) and (11), we have deg fik ≤ (n − 2)n0 ( n Y i=1 fi ) − (n − 1)(n − 2) . 2 (12) From (9) the same estimate as in (12) follows for deg fj . Thus the theorem is proved for such fj and fik . Inserting all the relations of the from (9) into the right side of equality fn = f1 + f2 + · · · + fn−1 , yields an equation of the form fr = κ1 fi1 + · · · + κq fiq , (13) where the κj ∈ F. Moreover, if one of these κν = 0, then the corresponding fiν must be appeared in one of the equations (9) with a nonzero λν . Hence, (12) is established for this fiν . Finally, for those κν 6= 0, we treat (13) exactly as we did (9), (note that q + 1 < n), and obtain the estimate (12) for deg fiν , and deg fn . This completes the induction in this case. Case II. f1 , f2 , · · · , fn−1 are linearly independent over F. By using Lemma 2, we have W [f1 , f2 , · · · , fn−1 ] 6= 0. Without lost of generality, we suppose that fn has the greatest degree, and therefore it is necessary to prove that deg fn ≤ (n − 2)n0 (f1 f2 · · · fn ) − (n − 1)(n − 2) . 2 Considering the equality fn = f1 + f2 + · · · + fn−1 , we have W [f1 , · · · , fn−2 , fn−1 ] = W [f1 , · · · , fn−2 , fn ]. It can be easily seen for any i (i = 1, · · · , n), (n−2) (fi , fi0 , · · · , fi ¯ ¯ )¯ W [f1 , · · · , fn−2 , fn−1 ]. (n−2) Since fi ’s are relatively prime by pairs, we conclude that (fi , fi0 · · · , fi prime. So, we get n ¯ Y (n−2) ¯ (fi , fi0 , · · · , fi )¯ W [f1 , · · · , fn−2 , fn−1 ]. )’s are relatively i=1 Now since W [f1 , · · · , fn−2 , fn−1 ] 6= 0, we conclude that n X (n−2) deg(fi , fi0 , · · · , fi ) ≤ deg W [f1 , · · · , fn−2 , fn−1 ]. i=1 Using the relations (1) and (2), we obtain n X (deg fi − (n − 2)n0 (fi )) ≤ deg(f1 f2 · · · fn−1 ) − i=1 or equivalently, (n − 1)(n − 2) 2 (n − 1)(n − 2) . 2 For proving (8), it is necessary to consider the case deg f1 = · · · = deg fn . Now the proof is clear using the relation (3). deg fn ≤ (n − 2)n0 (f1 f2 · · · fn ) − Vol. 1 87 An Extension of ABC-theorem Remark 2. In the case where the number of constant polynomials are more than one, the inequality (7) is not valid in general case. For example if f1 = · · · = f5 = 1, f6 = x and f7 = x + 5, then it is not true. Indeed, finding similar inequality for the case that constant polynomials are more than one is an open question yet. As an immediate result of the relation (7), we have: Corollary 1. With the assumption of the Theorem 1, we have deg(f1 f2 · · · fn ) ≤ n(n − 2)n0 (f1 f2 · · · fn ) − n(n − 1)(n − 2) . 2 Corollary 2. For n ≥ 3, suppose f1 , f2 , · · · , fn are non-constant and relatively prime by pairs. Then we obtain 1 n0 (f1 ) n0 (f2 ) n0 (fn ) < + + ··· + . n−2 deg f1 deg f2 deg fn Proof. Without loss of generality, we suppose that deg f1 ≤ · · · ≤ deg fn . Applying Theorem 1, yields deg fn < (n − 2)(n0 (f1 ) + · · · + n0 (fn )). Dividing the both sides of the above inequality by (n − 2) deg fn , completes the proof. §3. Application to the generalized Fermat-Catalan Equation Now, we deal with the generalized Fermat-Catalan equation [8]. Theorem 2. Consider the generalized Fermat-Catalan equation as follows m m2 n−1 mn 1 am 1 + a2 + · · · + an−1 = an , (14) in which a1 , a2 , · · · , an are elements of F[x] with char(F) = 0, such that they are relatively prime by pairs and at most one of ai ’s is constant. Then the equation (14) with condition n(n − 2) ≤ m = min {mi } has no solution in F[x]. 1≤i≤n m2 mn 1 Proof. Suppose f1 = am 1 , f2 = a2 , · · · , fn = an . These functions satisfy the conditions of Theorem 1. Thus we have m1 mn mn 1 m2 deg(am 1 a2 · · · an ) ≤ n(n − 2)n0 (a1 · · · an ) − n(n − 1)(n − 2) . 2 (15) We also have, mn 1 m2 m deg(a1 a2 · · · an ) ≤ deg(am 1 a2 · · · an ), (16) mn 1 m2 n0 (am 1 a2 · · · an ) = n0 (a1 a2 · · · an ) ≤ deg(a1 a2 · · · an ). (17) and Now considering the both relations (15)-(17), we get m deg(a1 a2 · · · an ) ≤ n(n − 2) deg(a1 a2 · · · an ) − n(n − 1)(n − 2) , 2 (18) 88 Morteza Bayat , Hossein Teimoori and Mehdi Hassani No. 2 or equivalently, (m − n(n − 2)) deg(a1 a2 · · · an ) ≤ − n(n − 1)(n − 2) . 2 (19) The last inequality result in m − n(n − 2) < 0, which is in contradiction with our theorem’s hypothesis. Therefore, we conclude that the Diophantine equation (14) has no solution in F[x]. Of course, there is in [10] a natural extension of the above result for several variables using the generalized Wronskian. Acknowledgment The authors thanks J. Browkin for his interesting suggestions about the abc-conjecture. References [1] W.Stothers, Polynomial identities and hauptmoduln, Quart. Math.Oxford, 32(1981), 349-370. [2] R.C.Mason, Diophantine equations over function fields, Londen Math. Soc. Lecture note series, Vol. 96, Cambridge University Press, 1984. [3] J.H. Silverman, The S-unit equation over function fields, Math. Proc. Cambridge PhiLos. Soc. 95(1984), No.1, 3-4. [4] S.Lang, Math Talks for Undergraduates, Spinger-Verlag 1999. [5] J.Oesterle, Nouvelles approches du “theorem” de Fermat. (New approches to Fermat’s last theorem ) Semin. Bourbaki, 40eme Annee, Vol. 1987/88, Exp. No. 694 Asterisque 161/162, 165-186(1988). [6] D.W.Masser, Note on a Conjecture of Szpiro. Les pinceaux de courbes elliptiques, semin., Paris/Fr. 1988, Asterisque 183, 19-23(1990). [7] C.L.Stewart and R.Tijdeman, On the Oesterle-Masser Conjecture, Monatshefte Math. 102(1986), 251-257. [8] L.N.Vaserstein, Quantum (abc)-Theorems, Journal of Number Theory, 81(2000), 351358. [9] H.N.Shapiro and G.H.Sparer, Extention of a Theorem of Mason, Comm. Pure and Appl. Math., 47(1994), 711-718. [10] M.Bayat and H.Teimoori, A new bound for an extension of Mason Theorem for functions of several variables, Archiv der Mathematik, 82 (2004), 230-239. Scientia Magna Vol. 1 (2005), No. 2, 89-90 An equation involving the Smarandache function Ma Jinping Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China,710069 Abstract For any positive integer n, let S(n) denotes the Smarandache function, and φ(n) is the Euler function. The main purpose of this paper is using the elementary method to study the solutions of the equation S(n) = φ(n), and give all solutions for it. Keywords Smrandache function; Equation; Solutions. §1. Introduction For any positive integer n, the Smarandache function S(n) is defined as the smallest integer m such that n|m!. From the definition and the properties of S(n), one can easily deduce that αk 1 α2 if n = pα 1 p2 · · · pk is the prime powers factorization of n, then i S(n) = max {S(pα i )}. 1≤i≤k About the arithmetical properities of S(n), many people had studied it before, see references [3], [4] and [5]. If n ≥ 1, the Euler function φ(n) is defined to be the number of all positive integers not exceeding n, which are relatively prime to n. It is clear that φ(n) is a multiplicative function. In this paper, we shall use the elementary method to study the solutions of the equation S(n) = φ(n), and give all solutions for it. That is, we shall prove the following: Theorem. The equation S(n) = φ(n) have only 4 solutions, namely, n = 1, 8, 9, 12. §2. Proof of the theorem αk 1 α2 In this section, we shall complete the proof of the theorem. Let n = pα 1 p2 · · · pk denotes the factorization of n into prime powers, and let S(n) = max {S(pi αi )} = S(pα ). 1≤i≤k Then from the definitions of S(n) and φ(n) we have φ(n) 1 −1 2 −1 k −1 = pα (p1 − 1)pα (p2 − 1) · · · pα (pk − 1) 1 2 k = φ(pα )φ(n1 ) = pα−1 (p − 1)φ(n1 ) = S(pα ). 90 Ma Jinping No. 2 It is clear that n = 1 is a solution of the equation S(n) = φ(n). If n > 1, then we will discuss the problem in three cases: (I) If α = 1 and n = p, then S(n) = p 6= p − 1 = φ(n). That is, there is no any prime satisfied the equation. If α = 1 and n = n1 p, then S(n) = p 6= (p − 1)φ(n1 ) = φ(n1 p). So the equation has also no solution. (II) If α = 2, then S(p2 ) = 2p and φ(p2 n1 ) = p(p − 1)φ(n1 ). So in this case S(n) = φ(n) if and only if (p − 1)φ(n1 ) = 2. This time, there are two cases: p − 1 = 1, φ(n1 ) = 2; p − 1 = 2, φ(n1 ) = 1. That is, p = 2, n1 = 3; p = 3, n1 = 1. So in this case, the equation has two solutions: n = 12, 9. (III) If α = 3, it is clear that S(23 ) = φ(23 ) = 4, so n = 8 satisfied the equation. If α ≥ 3 and p > 2, noting that α−2 pα−2 > 2α−2 = (1 + 1) = 1 + α − 2 + · · · + 1 > α. That is, pα−1 > αp ⇒ pα−1 (p − 1)φ(n1 ) > αp, but S(pα ) ≤ αp. So this time, the equation has no solution. Now combining the above three cases, we may immediately get all 4 solutions of equation S(n) = φ(n), namely n = 1, 8, 9, 12. This completes the proof of Theorem. References [1] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House, 1993. [2] Tom M. Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag, 1976. [3] Wang Yongxing, On the Smarandache function, Research on Smarandache problems in number theory, Hexis, 2005, pp. 103-106. [4]Ma Jinping, The Smaranache Multiplicative Function, Scientia Magna,1(2005), 125-128. [5] Li Hailong and Zhao Xiaopeng, On the Smarandache function and the K-th roots of a positive integer, Research on Smarandache problems in number theory, Hexis, 2004, pp. 119-122. Scientia Magna Vol. 1 (2005), No. 2, 91-95 Inequalities for the polygamma functions with application1 Chaoping Chen Department of Applied Mathmatics, Hennan Polytechnic University Jiaozuo, Hennan, P. R. China Abstract We present some inequalities for the polygamma funtions. As an application, we give n P 1 the upper and lower bounds for the expression − ln n − γ, where γ = 0.57721 · · · is the Euler’s k constant. Keywords k=1 Inequality; Polygamma function; Harmonic sequence; Euler’s constant. §1. Inequalities for the Polygamma Function The gamma function is usually defined for Rez > 0 by Z ∞ Γ(z) = tz−1 e−t dt. 0 The psi or digamma function, the logarithmic derivative of the gamma function and the polygamma functions can be expressed as ¶ 0 ∞ µ X 1 Γ (z) 1 = −γ + − , ψ(z) = Γ(z) 1+k z+k k=0 ψ n (z) = (−1)n+1 n! ∞ X k=0 1 (z + k)n+1 for Rez > 0 and n = 1, 2, · · · , where γ = 0.57721 · · · is the Euler’s constant. M. Merkle [2] established the inequality 2N ∞ 2N +1 X X X 1 1 B2k 1 1 B2k + 2+ < < + x 2x x2k+1 (x + k)2 x x2k+1 k=1 k=0 k=1 for all real x > 0 and all integers N ≥ 1, where Bk denotes Bernoulli numbers, defined by ∞ X Bj t = tj . t e − 1 j=0 j! The first five Bernoulli numbers with even indices are B2 = 1 This 1 1 1 1 5 , B4 = − , B6 = , B8 = − , B10 = . 6 30 42 30 66 work is supported in part by SF of Henan Innovation Talents at University of P. R. China 92 Chaoping Chen No. 2 The following theorem 1 establishes a more general result. Theorem 1. Let m ≥ 0 and n ≥ 1 be integers, then we have for x > 0, 2m+1 2m X B2j 1 X 1 B2j 1 1 − < ψ(x) < ln x − − ln x − 2j 2x 2j x 2x 2j x2j j=1 j=1 and (1) 2m X B2j Γ(n + 2j) (n − 1)! n! + n+1 + n x 2x (2j)! xn+2j j=1 < (−1)n+1 ψ (n) (x) < 2m+1 X B2j Γ(n + 2j) (n − 1)! n! + + . xn 2xn+1 (2j)! xn+2j j=1 (2) Proof. From Binet’s formula [6, p. 103] µ ¶ ¶ Z ∞µ √ 1 t t e−xt ln Γ(x) = x − ln x − x + ln 2π + − 1 + dt, 2 et − 1 2 t2 0 we conclude that 1 ψ(x) = ln x − − 2x Z 0 ∞ µ t t −1+ t e −1 2 and therefore n+1 (−1) ψ (n) (n − 1)! n! (n) = + n+1 + xn 2x Z ∞ µ 0 ¶ e−xt dt t t t −1+ et − 1 2 (3) ¶ tn−1 e−xt dt. (4) It follows from Problem 154 in Part I, Chapter 4, of [3] that 2m+1 2m X B2j X B2j 2j t t t < t −1+ < t2j (2j)! e − 1 2 (2j)! j=1 j=1 (5) for all integers m > 0. The inequality (5) can be also found in [4]. From (3) and (5) we conclude (1), and we obtain (2) from (4) and (5). This completes the proof of the theorem 1. Note that ψ(x + 1) = ψ(x) + x1 (see [1, p. 258]), (1) can be written as 2m+1 2m X B2j 1 X 1 1 B2j 1 − < ψ(x + 1) − ln x < − 2j 2x 2j x 2x 2j x2j j=1 j=1 (6) and (2) can be written as 2m X B2j Γ(n + 2j) n! (n − 1)! − n+1 + n x 2x (2j)! xn+2j j=1 n+1 < (−1) ψ (n) 2m+1 X B2j Γ(n + 2j) (n − 1)! n! (x) < − + . xn 2xn+1 (2j)! xn+2j j=1 (7) In particular, taking in (6) m = 0 we obtain for x > 0, 1 1 1 < ψ(x + 1) − ln x < − 2 2x 12x 2x (8) Vol. 1 93 Inequalities for the polygamma functions with application and taking in (7) m = 1 and n = 1, we obtain for x > 0 0 1 1 1 1 1 1 1 1 − 3+ − < − ψ (x + 1) < 2 − 3 + 2 5 7 2x 6x 30x 42x x 2x 6x 30x5 (9) The inequalities (8) and (9) play an important role in the proof of the theorem 2 in Section 2. §2. Inequalities for Euler’s Constant Euler’s constant γ = 0.57721 · · · is defined by µ ¶ 1 1 1 γ = lim 1 + + + · · · + − ln n . n→∞ 2 3 n It is of interest to investigate the bounds for the expression n P k=1 1 k − ln n − γ. The inequality n X1 1 1 1 − 2 < − ln n − γ < 2n 8n k 2n k=1 is called in literature Franel’s inequality [3, Ex. 18]. n−1 P 1 It is given in [1, p. 258] that ψ(n) = k − γ, and then we have get k=1 n X 1 − ln n − γ = ψ(n + 1) − ln n. k (10) k=1 Taking in (6) x = n we obtain that 2m+1 n 2m X B2j 1 X X 1 1 1 B2j 1 − < − ln n − γ < − . 2j 2n 2j n k 2n 2j n2j j=1 j=1 (11) k=1 The inequality (11) provides closer bounds for n P k=1 1 k − ln n − γ. L.Tóth [5, p. 264] proposed the following problems: (i) Prove that for every positive integer n we have 1 2n + 2 5 < n X 1 1 . − ln n − γ < k 2n + 31 k=1 (ii) Show that 25 can be replaced by a slightly smaller number, but that replaced by a slightly larger number. The following Theorem 2 answers the problem due to L.Tóth. Theorem 2. For every positive integer n, 1 3 can not be n X1 1 1 < − ln n − γ < , 2n + a i 2n + b i=1 with the best possible constants (12) 94 Chaoping Chen a= 1 −2 1−γ No. 2 and b= 1 3 Proof. By (10), the inequality (12) can be rearranged as 1 − 2n ≤ a. ψ(n + 1) − ln n b< Define for x > 0 φ(x) = 1 − 2x. ψ(x + 1) − ln x Differentiating φ and utilizing (8) and (9) reveals that for x > 12 5 0 1 − ψ (x + 1) − 2(ψ(x + 1) − ln x)2 x µ ¶2 1 1 1 1 1 12 − 5x < 2− 3+ − − 2 = < 0, 2x 6x 30x5 2x 12x2 360x5 0 (ψ(x + 1) − ln x)2 φ (x) = and then the function φ strictly decreases with x > Straightforward calculation produces φ(1) = φ(2) = φ(3) = 12 5 . 1 − 2 = 0.36527211862544155 · · · , 1−γ 3 2 1 − 4 = 0.35469600731465752 · · · , − γ − ln 2 11 6 1 − 6 = 0.34898948531361115 · · · . − γ − ln 3 Therefore, the sequence φ(n) = 1 − 2n, ψ(n + 1) − ln n n∈N is strictly decreasing. This leads to lim φ(n) < φ(n) ≤ φ(1) = n→∞ 1 − 2. 1−γ Making use of asymptotic formula of ψ (see [1, p. 259]) ψ(x) = ln x − 1 1 + O(x−4 ) − 2x 12x2 we conclude that lim φ(n) = lim φ(x) = lim n→∞ x→∞ This completes the proof of the theorem 2. x→∞ 1 3 (x → ∞), + O(x−2 ) 1 = . 1 + O(x−1 ) 3 Vol. 1 Inequalities for the polygamma functions with application 95 References [1] M. Abramowitz and I.Stegun (Eds), Handbook of Mathematical Function with Formulas, Graphs, and Mathematical Tables, 4th printing, with corrections, Applied Mathematics Series 55, National Bureau of Standards, Washington, 1965. [2] M. Merkle, Logarithmic convexity inequalities for the gemma function, J. Math. Anal. Appl, 203(1996), 369–380. [3] G. Pólya and G. Szegö, Problems and Theorems in analysis, Vol.I and II, SpringerVerlag, Berlin, Heidelberg, 1972. [4] Z. Sasvári, Inequalities for binomial coefficients, J. Math. Anal. Appl. 236(1999), 223-226. [5] L. Tóth, E 3432, Amer. Math. Monthly 98(1991), 264; 99(1992), 684-685. [6] Zh-x.Wang and D.-R.Guo, Introduction to Special Function, the Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000(Chinese). Scientia Magna Vol. 1 (2005), No. 2, 96-108 On the number of Smarandache zero-divisors and Smarandache weak zero-divisors in loop rings W.B.Vasantha and Moon K.Chetry Department of Mathematics I.I.T.Madras,Chennai Abstract In this paper we find the number of smarandache zero divisors (S-zero divisors) and smarandache weak zero divisors (S-weak zero divisors) for the loop rings Z2 Ln (m) of the loops Ln (m) over Z2 . We obtain the exact number of S-zero divisors and S-weak zero divisors when n = p2 or p3 or pq where p, q are odd primes. We also prove ZLn (m) has infinitely many S-zero divisors and S-weak zero divisors, where Z is the ring of integers. For any loop L we give conditions on L so that the loop ring Z2 L has S-zero divisors and S-weak zero divisors. §0 . Introduction This paper has four sections. In the first section, we just recall the definitions of Szero divisors and S-weak zero divisors and some of the properties of the new class of loops Ln (m). In section two, we obtain the number of S-zero divisors of the loop rings Z2 Ln (m) p−1 X p+1 and show when n = p2 , where p is an odd prime, Z2 Ln (m) has p(1 + Cr ) S-zero r=2, r even 2 pX −1 divisors. Also when n = p3 , p an odd prime, Z2 Ln (m) has p(1 + p2 +1 Cr ) + p2 (1 + r=2, r even p−1 X p+1 Cr ) S-zero divisors. Again when n = pq, where p, q are odd primes, Z2 Ln (m) has r=2,reven p + q + p( q−1 X q+1 Cr ) + q( r=2, r even p−1 X p+1 Cr ) S-zero divisors. Further we prove ZLn (m) has r=2, r even infinitely many S-zero divisors. In section three, we find the number of S-weak zero divisors for the loop ring Z2 Ln (m) and prove that when n = p2 , where p is an odd prime, Z2 Ln (m) p−1 X p+1 has 2p(1 + Cr ) S-weak zero divisors. Also when n = p3 , where p is an odd prime, r=2, r even Z2 Ln (m) has 2p( 2 pX −1 r=2,reven p2 +1 Cr ) + 2p2 ( p−1 X p+1 Cr ) S-weak zero divisors. Again when r=2, r even n = pq, where p, q are odd primes, Z2 Ln (m) has 2[p( q−1 X r=2, r even q+1 Cr ) + q( p−1 X p+1 Cr )] r=2, r even S-weak zero divisors. We prove ZLn (m) has infinitely many S-weak zero divisors. The final section gives some unsolved problems and some conclusions based on our study. Vol. 1 97 On the number of Smarandache Zero-Divisors and Weak Zero-divisors §1. Basic Results Here we just recollect some basic results to make this paper a self contained one. Definition 1.1[4]. Let R be a ring. An element a ∈ R\{0} is said to be a S-zero divisor if a.b = 0 for some b 6= 0 in R and there exists x, y ∈ R\{0, a, b} such that i. a.x = 0 or x.a = 0 ii b.y = 0 or y.b = 0 iii. x.y 6= 0 or y.x 6= 0 Definition 1.2[4]. Let R be a ring. An element a ∈ R\{0} is a S-weak zero divisor if there exists b ∈ R\{0, a} such that a, b = 0 satisfying the following conditions: There exists x, y ∈ R\{0, a, b} such that i. a.x = 0 or x.a = 0 ii. b.y = 0 iii. x.y = 0 or y.b = 0 or y.x = 0 Definition 1.3[3]. Let Ln (m) = {e, 1, 2, 3 · · · , n} be a set where n > 3, n is odd and m is a positive integer such that (m, n) = 1 and (m − 1, n) = 1 with m < n. Define on Ln (m), a binary operation 0 .0 as follows: i. e.i = i.e f or ii. i2 . = e iii. i.j = t, where all f or i ∈ Ln (m)\{e} all i ∈ Ln (m) t ≡ (mj−(m−1)i)(mod n) f or all i, j ∈ Ln (m), i 6= e and j 6= e. Then Ln (m) is a loop. This loop is always of even order; further for varying m, we get a class of loops of order n + 1 which we denote by Ln . Example 1.1[3]. Consider L5 (2) = {e, 1, 2, 3, 4, 5}. The composition table for L5 (2) is given below: . e 1 2 3 4 5 e e 1 2 3 4 5 1 1 e 3 5 2 4 2 2 5 e 4 1 3 3 3 4 1 e 5 2 4 4 3 5 2 e 1 5 5 2 4 1 3 e This loop is non-commutative and non-associative and of order 6. Theorem 1.1[3]. Let Ln (m) ∈ Ln . For every t|n there exists t subloops of order k + 1, where k = n/t. 98 W.B.Vasantha and Moon K.Chetry No. 2 Theorem 1.2[3]. Let Ln (m) ∈ Ln . If H is a subloop of Ln (m) of order t + 1, then t|n. Remark 1.2[3]. Lagrange’s theorem is not satisfied by all subloops of the loop Ln (m),i.e there always exists a subloop H of Ln (m) which does not satisfy the Lagrange’s theorem, i.e o(H) † o(Ln (m)). §2. Definition of the number of S-zero divisors in Z2 Ln (m) and ZLn (m) In this section, we give the number of S-zero divisors in Z2 Ln (m). We prove ZLn (m) (where n = p2 or pq, p and q are odd primes), has infinitely many S-zero divisors. Further we show any loop L of odd (or even) order if it has a proper subloop of even (or odd) order then the loop ring Z2 Ln (m) over the field Z2 has S-zero divisors. We first show if L is a loop of odd order and L has a proper subloop of even order, then Z2 Ln (m) has S-zero divisors. Theorem 2.1. Let L be a finite loop of odd order. Z2 = {0, 1}, the prime field of characteristic 2. Suppose H is a subloop of L of even order, then Z2 L has S-zero divisors. Proof. Let |L| = n; where n is odd. Z2 L be the loop ring of L over Z2 . H be the subloop m n X X hi , then gi and Y = of L of order m, where m is even. Let X = i=1 i=1 X.Y = 0. Now (1 + gt )X = 0, gt ∈ l\H. also (1 + hi + hj + hk )Y = 0, hi , hj , hk ∈ H. so that (1 + gt )(1 + hi + hj + hk ) 6= 0. Hence the claim. Corollary 2.1. If L is a finite loop of even order n and H is a subloop of odd order m, then the loop ring Z2 L has S-zero divisors. It is important here to mention that Z2 L may have other types of S-zero divisors. This theorem only gives one of the basic conditions for Z2 L to have S-zero divisors. Example 2.1. Let Z2 L25 (m) be the loop ring of the loop L25 (m) over Z2 , where (m, 25) = 1 and (m − 1, 25) = 1. As 5|25, so L25 (m) has 5 proper subloops each of order 6. Let H be one of the proper subloops of L25 (m). Now take 26 6 X X X= gi , Y = hi , gi ∈ L25 (m), hi ∈ H, i=1 i=1 then (1 + gi )X = 0, gi ∈ L25 (m)\H Vol. 1 On the number of Smarandache Zero-Divisors and Weak Zero-divisors (1 + hi )Y = 0, 99 hi ∈ H but (1 + gi )(1 + hi ) 6= 0. so X and Y are S-zero divisors in Z2 L25 (m). Theorem 2.2. Let Ln (m) be a loop of order n + 1 (n an odd number,n > 3) with n = p2 , p an odd prime. Z2 be the prime field of characteristic 2. The loop ring Z2 Ln (m) has exactly à ! p−1 X p+1 p 1+ Cr r=2, r even S-zero divisors. Proof. Given Ln (m) is a loop of order n+1, where n = p2 (p an odd prime). Let Z2 Ln (m) be the loop ring of the loop Ln (m) over Z2 . Now clearly the loop Ln (m) has exactly p subloops of order p + 1. The number of S-zero divisors in Z2 Ln (m) for n = p2 can be enumerated in the following way: Let p+1 n+1 X X X= gi and Y = hi i=1 i=1 where gi ∈ Ln (m) and hi ∈ Hj . For this X.Y = 0 choose a = (1 + g), g ∈ Ln (m)\Hj b = (hi + hj ), h i , h j ∈ Hj then a.X = 0 and b.Y = 0 but a.b 6= 0. So X and Y are S-zero divisors. There are p such S-zero divisors, as we have p subloops Hj (j = 1, 2, · · · , p) of Ln (m). Next consider, S-zero divisors of the form (h1 + h2 ) n+1 X gi = 0, where h1 , h2 ∈ Hj , gi ∈ Ln (m) i=1 put X = (h1 + h2 ), Y = n+1 X gi i=1 we have p+1 C2 such S-zero divisors. This is true for each of the subloops. Hence there exists p+1 C2 × p such S-zero divisors. Taking four elements h1 , h2 , h3 , h4 from Hj at a time, we get (h1 + h2 + h3 + h4 ) n+1 X i=1 gi = 0 100 so we get W.B.Vasantha and Moon K.Chetry p+1 No. 2 C4 × p such S-zero divisors. Continue in this way, we get (h1 + h2 + · · · + hp−1 ) n+1 X gi = 0, where h1 , h2 , · · · , hp−1 ∈ Hj i=1 So we get p+1 Cp−1 × p such S-zero divisors. Adding all these S-zero divisors, we get à ! p−1 X p+1 p 1+ Cr r=2, r even number of S-zero divisors in the loop ring Z2 Ln (m). Hence the claim. Example 2.2. Let Z2 L49 (m) be the loop ring of the loop L49 (m) over Z2 , where (m, 49) = 1 and (m − 1, 49) = 1. Here p = 7, so from Theorem 2.2, Z2 L49 (m) has à ! 6 X 7+1 7 1+ Cr r=2, r even 6 X S-zero divisors i.e 7(1 + 8 Cr ) = 889 S-zero divisors. r=2, r even Theorem 2.3. Let Ln (m) be a loop of order n + 1 (n an odd number, n > 3) with n = p3 , p an odd prime. Z2 be the prime field of characteristic 2. The loop ring Z2 Ln (m) has exactly   à ! 2 pX −1 p−1 X 2 p +1 2 p+1   p 1+ Cr + p 1 + Cr r=2, r even r=2, r even S-zero divisors. Proof. We enumerate all the S-zero divisors of Z2 Ln (m) in the following way: Case I: As p|p3 , Ln (m) has p proper subloops Hj each of order p2 + 1. In this case I, we have p2 − 1 types of S-zero divisors. We just index them by type I1 , type I2 , · · · , type Ip2 −1 . Type I1 : Here n+1 X i=1 gi 2 pX +1 hi = 0, gi ∈ Ln (m), hi ∈ Hj , (j = 1, 2, · · · , p) i=1 So we will get p S-zero divisors of this type. Type I2 : n+1 X (h1 + h2 ) gi = 0, h1 , h2 ∈ Hj (j = 1, 2, · · · , p). i=1 As in the Theorem 2.2, we will get Type I3 : (h1 + h2 + h3 + h4 ) p2 +1 n+1 X C2 × p S-zero divisors of this type. gi = 0, h1 , h2 , h3 , h4 ∈ Hj (j = 1, 2, · · · , p). i=1 2 We will get p +1 C4 × p S-zero divisors of this type. Continue this way, Vol. 1 On the number of Smarandache Zero-Divisors and Weak Zero-divisors 101 Type Ip2 −1 : (h1 + h2 + · · · + hp2 −1 ) n+1 X gi = 0, h i ∈ Hj i=1 2 We will get p +1 Cp2 −1 × p S-zero divisors of this type. Hence adding all this types of S-zero divisors we will get   2 pX −1 2 p +1 p 1 + Cr  r=2, r even S-zero divisors for case I. Case II: Again p2 |p3 , so there are p2 subloops Hj each of order p + 1. Now we can enumerate all the S-zero divisors in this case exactly as in case I above. So there are p−1 X p2 (1 + p+1 Cr ) r=2, r even S-zero divisors. Hence the total number of S-zero divisors in Z2 Ln (m) is   à ! 2 pX −1 p−1 X 2 p +1 2 p+1 p 1 + Cr  + p 1 + Cr r=2, r even r=2, r even Hence the claim. Example 2.3. Let Z2 L27 (m) be the loop ring of the loop L27 (m) over Z2 , where (m, 27) = 1 and (m − 1, 27) = 1. Here p = 3, so from Theorem 2.3, Z2 L27 (m) has 8 X 3(1 + 32 +1 Cr ) + 3 (1 + r=2, r even à S-zero divisors i.e 3 1 + 8 X 4 Cr ) r=2, r even ! 10 2 X 2 Cr r=2, r even à +9 1+ ! 2 X 4 Cr = 1533 S-zero divisors. r=2, r even Theorem 2.4. Let Ln (m) be a loop of order n + 1 (n an odd number, n > 3) with n = pq, where p, q are odd primes. Z2 be the prime field of characteristic 2. The loop ring Z2 Ln (m) has exactly à ! à ! q−1 p−1 X X q+1 p+1 p+q+p 1+ Cr + q 1 + Cr r=2, r even r=2, r even S-zero divisors. Proof. We will enumerate all the S-zero divisors in the following way: Case I: As p|pq, Ln (m) has p subloops Hj each of order q + 1. Proceeding exactly in the à ! q−1 X q+1 same way as in the Theorem 2.3, we will get p + p 1 + Cr S-zero divisors for r=2, r even case I. Case II: Again q|pq, so Ln (m) has q subloops Hj each of order p + 1. Now as above we à ! p−1 X p+1 will get q + q 1 + Cr S-zero divisors for case II. Hence adding all the S-zero r=2, r even 102 W.B.Vasantha and Moon K.Chetry divisors in case I and case II, we get à q−1 X p+q+p 1+ ! q+1 Cr à ! p−1 X +q 1+ r=2, r even No. 2 p+1 Cr r=2, r even S-zero divisors in Z2 Ln (m). Hence the claim. Now we prove for the loop ring ZLn (m) when n = p2 or p3 or pq, where p, q are odd primes, ZLn (m) has infinitely many S-zero divisors. Theorem 2.5. Let ZLn (m) be the loop ring of the loop Ln (m) over Z, where n = p2 or p3 or pq (p, q are odd primes), then ZLn (m) has infinitely many S-zero divisors. Proof. Let Ln (m) be a loop ring such that n = p2 . Ln (M ) has p subloops (say Hj ) each of order p + 1. Now the loop ring ZLn (m) has the following types of S-zero divisors: X = a − bh1 + bh2 − ah3 and Y = n+1 X gi i=1 where a, b ∈ Z and hi ∈ Hi , gi ∈ Ln (m) such that (a − bh1 + bh2 − ah3 ) n+1 X gi = 0 i=1 Again (1 − gk )Y = 0, gk ∈ Ln (m)\Hj also (a − bh1 + bh2 − ah3 ) clearly X  (1 − gk )  hi = 0, h i ∈ Hj  X hi  6= 0. hi ∈Hj So X, Y are S-zero divisors in ZLn (m). Now we see there are infinitely many S-zero divisors of this type for a and b can take infinite number of values in Z. For n = p2 or p3 or pq we can prove the results in a similar way. Hence the claim. §3. Determination of the number of S-weak zero divisors in Z2 Ln (m) and ZLn (m) In this section, we give the number of S-weak zero divisors in the loop ring Z2 Ln (m) when n is of the form p2 , p3 or pq where p and q are odd primes. Before that we prove the existence of S-weak zero divisors in the loop ring Z2 L whenever L has a proper subloop. Theorem 3.1. Let n be a finite loop of odd order. Suppose H is a subloop of L of even order, then Z2 L has S-weak zero divisors. Vol. 1 On the number of Smarandache Zero-Divisors and Weak Zero-divisors 103 Proof. Let |L| = n; n odd. Z2 L be the loop ring. H be the subloop of L of order m, n X where m is even. Let X = gi and Y = 1 + ht , gi ∈ L, ht ∈ H, then i=1 X.Y = 0 Now Y. m X hi = 0, hi ∈ H i=1 also X(1 + gt ) = 0, so that (1 + gt ) gt (6= ht ) ∈ H m X hi = 0. i=1 Hence the claim. Example 3.1. Let Z2 L25 (m) be the loop ring of the loop L25 (m) over Z2 , where (m, 25) = 1 and (m − 1, 25) = 1. As 5|25, so L25 (m) has 5 proper subloops each of order 6. Take 26 X X= gi , Y = 1 + ht , gi ∈ L25 (m), ht ∈ H i=1 then X.Y = 0 again X(1 + gt ) = 0, Y 6 X gt (6= ht ) ∈ H hi = 0, hi ∈ H i=1 also (1 + gt ) 6 X hi = 0, i=1 So X and Y are S-weak zero divisors in Z2 L25 (m). Example 3.2. Let Z2 L21 (m) be the loop ring of the loop L21 (m) over Z2 , where where (m, 21) = 1 and (m − 1, 21) = 1. As 3|21, so L21 (m) has 3 proper subloops each of order 8. Take 8 X X= hi , Y = 1 + ht , h i , h t ∈ H i=1 then X.Y = 0 again X(1 + gt ) = 0, Y 22 X i=1 gi = 0, gt (6= ht ) ∈ H gi ∈ L21 (m) 104 W.B.Vasantha and Moon K.Chetry also (1 + gt ) 22 X No. 2 gi = 0, i=1 So X and Y are S-weak zero divisors in Z2 L21 (m). Theorem 3.2. Let Ln (m) be a loop of order n + 1 (n an odd number, n > 3) with n = p2 , p an odd prime. Z2 be the prime field of characteristic 2. The loop ring Z2 Ln (m) has exactly à p−1 ! X p+1 2p Cr r=2, r even S-weak zero divisors. Proof. Clearly the loop Ln (m) has p subloops Hj each of order p + 1. As in case of Theorem 2.3, we index the p − 1 types of S-weak zero divisors by I1 , I2 , · · · , Ip−1 . Now the number of S-weak zero divisors in Z2 Ln (m) for n = p2 can be enumerated in the following way: Type I1 . Let n+1 X gi X = h1 + h2 , Y = i=1 where h1 , h2 ∈ Hj and gi ∈ Ln (m) then XY = 0 take a= p+1 X hi , and b = h3 + h4 where hi ∈ Hj , (j = 1, 2, · · · , p) i=1 then aX = 0, bY = 0 also ab = 0 So for each proper subloop we will get p+1 C2 S-weak zero divisors and as there are p proper subloops we will get p+1 C2 × p such S-weak zero divisors. Type I2 . Again let p+1 X X = h1 + h2 , Y = hi , h i ∈ H j i=1 then XY = 0 take a= n+1 X gi , gi ∈ Ln (m), b = h1 + h2 , i=1 then aX = 0, bY = 0 h 1 , h 2 ∈ Hj , Vol. 1 On the number of Smarandache Zero-Divisors and Weak Zero-divisors 105 also ab = 0 Here also we will get Type I3 . p+1 C2 × p such S-weak zero divisors of this type. (h1 + h2 + h3 + h4 ) n+1 X gi , gi ∈ Ln (m), h i ∈ Hj . i=1 As above we can say there are Type I4 . p+1 C4 × p such S-weak zero divisors. (h1 + h2 + h3 + h4 ) p+1 X hi , h i ∈ Hj . i=1 There are p+1 C4 × p such S-weak zero divisors. Continue this way, Type Ip−2 . (h1 + h2 + · · · + hp−1 ) n+1 X gi , gi ∈ Ln (m), h i ∈ Hj . i=1 there are p+1 Cp−1 × p such S-weak zero divisors. Type Ip−1 . (h1 + h2 + · · · + hp−1 ) n X hi , h i ∈ Hj . i=1 Again there are p+1 Cp−1 × p such S-weak zero divisors of this type. Adding all these S-weak zero divisors we will get the total number of S-weak zero divisors in Z2 Ln (m) as à p−1 ! X p+1 2p Cr r=2, r even Hence the claim. Theorem 3.3. Let Ln (m) be a loop of order n + 1 (n an odd number, n > 3) with n = p3 , p an odd prime. Z2 be the prime field of characteristic 2. The loop ring Z2 Ln (m) has exactly   à p−1 ! 2 pX −1 X 2 p +1 2 p+1   2p Cr + 2p Cr r=2, r even r=2, r even S-weak zero divisors. Proof. We enumerate all the S-weak zero divisors of Z2 Ln (m) in the following way: Case I: As p|p3 , Ln (m) has p proper subloops Hj each of order p2 + 1. Now as in the Theorem 3.2. Type I1 : n+1 X (h1 + h2 ) gi = 0, gi ∈ Ln (m), hi ∈ Hj . i=1 So we will get p2 +1 C2 × p S-weak zero divisors of type I1 . 106 W.B.Vasantha and Moon K.Chetry No. 2 Type I2 : (h1 + h2 ) 2 pX +1 hi = 0, h i ∈ Hj . i=1 2 So we will get p +1 C2 × p S-weak zero divisors of type I2 . Continue in this way Type Ip2 −2 : n+1 X (h1 + h2 + · · · + hp2 −1 ) gi = 0, i=1 2 So we will get p +1 Cp2 −1 × p S-weak zero divisors of this type. Type Ip2 −1 : 2 pX +1 (h1 + h2 + · · · + hp2 −1 ) hi = 0, i=1 p2 +1 So we will get Cp2 −1 × p S-weak zero divisors of type Ip2 −1 . Adding all this S-weak zero we willget the total number of S-weak zero divisors  divisors, 2 pX −1 p2 +1 (in case I) in Z2 Ln (m) as 2p  Cr . 2 3 r=2, r even 2 Case II: Again p |p , so there are p proper subloops Hj each of order p + 1. Now we can enumerate all the S-weak zero divisors in this case exactly as in case I above. So there are à p−1 ! X 2 p+1 2p Cr r=2, r even S-weak zero divisors in case II. Hence the total number of S-weak zero divisors in Z2 Ln (m) is   à p−1 ! 2 pX −1 X 2 p +1 2 p+1 2p  Cr  + 2p Cr r=2, r even r=2, r even Hence the claim. Theorem 3.4. Let Ln (m) be a loop of order n + 1 (n an odd number, n > 3) with n = pq, p, q are odd primes. Z2 be the prime field of characteristic 2. The loop ring Z2 Ln (m) has exactly " à q−1 ! à p−1 !# X X q+1 p+1 2 p Cr + q Cr r=2, r even r=2, r even S-weak zero divisors. Proof. We will enumerate all the S-weak zero divisors in the following way: Case I: As p|pq, Ln (m) has p proper subloops Hj each of order q + 1. Proceeding exactly same way as in Theorem 3.3, we will get à q−1 ! X q+1 2p Cr r=2, r even Vol. 1 On the number of Smarandache Zero-Divisors and Weak Zero-divisors 107 S-weak zero divisors in case I. Case II: Again as q|pq, Ln (m) has q proper subloops Hj each of order p + 1. So as above we will get à p−1 ! X p+1 2q Cr r=2, r even S-weak zero divisors in case II. Hence adding all the S-weak zero divisors in case I and case II, we get " à 2 p ! q−1 X q+1 Cr à p−1 X +q r=2, r even !# p+1 Cr r=2, r4 even S-weak zero divisors in Z2 Ln (m). Hence the claim. Now we prove for the loop ring ZLn (m) where n = p2 or p3 or pq, (p, q are odd primes), ZLn (m) has infinitely many S-weak zero divisors. Theorem 3.5. Let ZLn (m) be the loop ring of the loop Ln (m) over Z, where n = p2 or p or pq (p, q are odd primes), then ZLn (m) has infinitely many S-weak zero divisors. 3 Proof. Let Ln (m) be a loop ring such that n = p2 . Ln (M ) has p subloops (say Hj ) each of order p + 1. Now the loop ring ZLn (m) has the following types of S-weak zero divisors: X = a − bh1 + bh2 − ah3 and Y = n+1 X gi i=1 where a, b ∈ Z, gi ∈ Ln (m) and h1 , h2 , h3 ∈ Hj are such that XY = 0. Again X p+1 X hi = 0, h i ∈ Hj i=1 also (1 − gt )Y = 0, clearly (1 − gt ) gt (6= ht ) ∈ Hj Ãp+1 X ! hi = 0. i=1 So X, Y are S-weak zero divisors in ZLn (m). Now we see there are infinitely many S-weak zero divisors of this type for a and b can take infinite number of values in Z. For n = p2 or p3 or pq we can prove the results in a similar way. Hence the claim. 108 W.B.Vasantha and Moon K.Chetry No. 2 §4. Conclusions: In this paper we find the exact number of S-zero divisors and S-weak zero divisors for the loop rings Z2 Ln (m) in case of the special type of loops Ln (m) ∈ Ln over Z2 , when n = p2 or p3 or pq (p, q are odd primes). We also prove for the loop ring ZLn (m) has infinite number of S-zero divisors and S-weak zero divisors. We obtain conditions for any loop L to have S-zero divisors and S-weak zero divisors. We suggest it would be possible to enumerate in the similar way the number of S-zero divisors and S-weak zero divisors for the loop ring Z2 Ln (m) when n = ps , s > 3; p a prime or when p = p1 p2 · · · pt where p1 , p2 , · · · , pt are odd primes. However we find it difficult when we take Zp instead of Z2 , where p can be odd prime or a composite number such that (p, n + 1 = 1) or (p, n + 1 = p) and n is of the form n = pt11 pt22 · · · ptrr , ti > 1, n is odd and p1 , p2 , · · · pr are odd primes. References [1] [2] [3] [4] R.H. Bruck, A survey of binary system, Spinger Verlag (1958). D.S.Passman, The algebraic structure of group rings, Wiley interscience, (1977). S.V.Singh, On a new class of loops and loop rings, PhD thesis, IIT Madras, (1994). Vasantha Kandasamy,W.B, Smarandache Zero divisors, (2001). http://www.gallup.unm.edu/smarandache/Zero-divisor.pdf Scientia Magna Vol. 1 (2005), No. 2, 109-110 The function equation S(n) = Z(n) 1 Maohua Le Department of Mathematics, Zhanjiang Normal College 29 Cunjin Road, Chikan Zhanjiang, Guangdong, P. R. China Abstract For any positive integer n, let S(n) and Z(n) denote the Smarandache function and the pseudo Smarandache function respectively. In this paper we prove that the equation S(n) = Z(n) has infinitely many positive integer solutions n. Keywords Smarandache function; Pseudo Smarandache function; Diophantine equation. For any positive integers n, let S(n) and Z(n) denote the Smarandache function and pseudo Smarandache function respectively. In [1], Ashbacher proposed two problems concerning the equation S(n) = Z(n) (1) as follows. Problem 1. Prove that if n is an even perfect number, then n satisfies (1). Problem 2. Prove that (1) has infinitely many positive integer solutions n. In this paper we completely solve these problems as follows. Theorem 1. If n is an even perfect number, then (1) holds. Theorem 2. (1) has infinitely many positive integer solutions n. Proof of Theorem 1. By [2, Theorem 277], if n is an even perfect number, then n = 2p−1 (2p − 1), (2) where p is a prime. By [3] and [4], we have S(n) = 2p − 1. (3) 1 p (2 − 1) ((2p − 1) + 1) = n, 2 (4) Z(n) = 2p − 1 (5) On the other hand, since by (2), we get immediately. The combination of (3) and (5) yields (1). Thus, the theorem is proved. This work is supported by N.S.F. of P. R. China(10271104), the Guangdong Provincial Natural Science and the Natural Science Foundation of the Education Department of Guangdong Province(0161). 1 Foundation(011781) 110 Maohua Le Proof of Theorem 2. S(p) = p, we have No. 2 Let p be an odd prime with p ≡ 3( mod 4). Since S(2) = 2 and S(2p) = max(S(2), S(p)) = max(2, p) = p. (6) Let t = Z(2p), By the define of Z(n), we have 1 t(t + 1) ≡ 0(mod2p). 2 (7) It implies that either t ≡ 0(modp) or t + 1 ≡ 0(modp). Hence, we get t ≥ p − 1. If t = p − 1, then from (7) we obtain 1 (p − 1)p ≡ 0(mod2p). 2 (8) 1 (p − 1)p ≡ 0(mod2). 2 (9) whence we get But, since p ≡ 3(mod4), (9) is impossible. So we have t ≥ p. (10) 1 p(p + 1) ≡ 0(mod2p) 2 (11) Since p + 1 ≡ 0(mod4), we get and t = p by (10). Therefore, by (6), n = 2p is a solution of (1). Notice that there exist infinitely many primes p with p ≡ 3(mod4). It implies that (1) has infinitely many positive integer solutions n. The theorem is proved. References [1] C.Ashbacher, Problems, Smrandache Notions J. 9(1998), 141-151. [2] G.H.Hardy and E.M.Wright, An introduction to the theory of numbers, Oxford University Press, Oxford, 1938. Scientia Magna Vol. 1 (2005), No. 2, 111-113 On the Smarandache Pseudo-number Sequences Li Zhanhu†,‡ † Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China ‡ Department of Mathematics, Xianyang Teacher’s College Xianyang, Shannxi, P.R.China Abstract The main purpose of this paper is using elementary method to study the main value of the m-th power mean of the sum of all digits in the Smarandache pseudo-number sequence, and give some interesting asymptotic formulae for them. Keywords Smarandache Pseudo-multiple of 5, pseudo-even, pseudo-odd sequence number; Sum of digits; Asymptotic formulae. §1. Introdution A number is called Smarandache pseudo-multiple of 5 if some permutation of the digits is a multiple of 5, including the identity permutation. For example: 51, 52, 53, 54, 56, 57, 58, 59, 101, 102, 103, 104, 106 · · · are Smarandache pseudo-multiple of 5 numbers. Similarly we can define the Smarandache pseudo-even numbers and the Smarandache pseudo-odd numbers. In reference [1], Professor F.Smarandache asked us to study the properties of the pseudo-multiple of 5, pseudo-even, pseudo-odd sequence. Let A denote the set of all Smarandache Pseudo-multiple of 5 numbers; Let B denote the set of all Smarandache Pseudo-even numbers and Let C denote the set of all Smarandache Pseudo-odd numbers. For convenience, denoted by A(n), the sum of all the digits of the base 10 digits of n. That is A(n) = k X ai i=0 if n = ak 10k + ak−1 10k−1 + · · · + a1 10 + a0 . In this paper, we shall use the element method to study the mean value of the m-power of the sum of all digits in the pseudo-number sequence, and give some interesting formulae for them. That is, we shall prove the following results: Theorem 1. For any integer number x ≥ 1, we have the asymptotic formula µ ¶m X ¡ ¢ 9 m A (n) = x + O x(log x)m−1 . log x 2 n∈A n≤x Theorem 2 For any integer number x ≥ 1, we have the asymptotic formula µ ¶m X 9 Am (n) = x log x + O(x(log x)m−1 ). 2 n∈B n≤x 112 Li Zhanhu Theorem 3 No. 2 For any integer number x ≥ 1, we have the asymptotic formula ¶m µ X 9 log x + O(x(log x)m−1 ). Am (n) = x 2 n∈C n≤x §2. Some lemmas To complete the proof of the theorem, we need the following lemmas. Lemma 1. For any integer number x ≥ 1, we have the asymptotic formula µ ¶m X ¡ ¢ 9 m A (n) = x log x + O x(log x)m−1 . 2 n≤x Proof. See reference [1]. Lemma 2. For any integer number x ≥ 1. Let D denotes the complementary set of A, then we have the asymptotic formula µ ¶ X (log x)m Am (n) = O x 5 log x . (4) n∈D n≤x Proof. From the definition of the set D, we know that the base 10 digits of the numbers in D are 1, 2, 3, 4, 6, 7, 8, 9, not including 0, 5. So, there are 8m m-digit number in D. Hence, for any integer n, there is a k such that 10k−1 ≤ x < 10k . Then we have X Am (n) ≤ X Am (n) t=1 10t−1 ≤n<10t n∈D n∈D n≤x Noting that k X X Am (n) < (9t)m × 8t , 10t−1 ≤n<10t n∈D we can write k X X Am (n) < t=1 10t−1 ≤n<10t n∈D k X (9t)m × 8t < 9m × k m × 8k+1 . t=1 Since k ≤ (log x) + 1 < k + 1, we have X ¡ m m log x A (n) = O (log x) × 8 n∈D n≤x ¢ µ (log x)m = O x 5 log x (4) ¶ . This proves Lemma 2. Lemma 3. For any integer number x ≥ 1. Let E denote the complementary set of B, then we have the asymptotic formula µ ¶ X (log x)m Am (n) = O x log x . 2 n∈E n≤x Vol. 1 Proof. On the Smarandache pseudo-number Sequences 113 By use the same method of proving Lemma 2, we can also get this Lemma. §3. Proof of the theorems Now we complete the proof of the theorems. First we prove Theorem 1. From the definition of Smarandache pseudo-multiple of 5 numbers, Lemma 1 and Lemma 2, we can get X X X Am (n) = Am (n) − Am (n) n∈A n≤x n≤x µ n∈D n≤x ¶m µ ¶ 9 (log x)m m−1 = x log x + O(x(log x) ) − O x 5 log x 2 (4) ¶m µ 9 log x + O(x(log x)m−1 ). = x 2 This completes the proof of Theorem 1. Using the same method of proving Theorem 1, we can also deduce the other Theorems. References [1] F. Smarandache, Only problem, Not Solution, Chicago, Xiquan Publ. House, 1993. [2] Harald Riede, Asymptotic estimation of a sum of digits, Fibonacci Quarterly, 36(1)(1998), 72-75. Scientia Magna Vol. 1 (2005), No. 2, 114-115 The classical Smarandache function and a formula for twin primes Dhananjay P.Mehendale Department of Electronics, S.P.College, Pune, India Abstract This short paper presents an exact formula for counting twin prime pairs less than or equal to x in terms of the classical Smarandache Function. An extension of the formula to count prime pairs (p, p + 2n), n > 1 and a positive interger is also given. §1. Introduction The most known Smarandache function which has become a classical Smarandache function in number theory is defined as follows: Definition. The classical Smarandache function, S, is a function S : N → N, N , the set of natural numbers such that S(1) = 1, and S(n)=The smallest integer such that n/S(n)!. This function has been extensively studied and many interesting properties of it have been discovered [1]. Subsequently many Smarandache type functions have been defined and their interesting properties have been achieved. Ruiz and Perez have discussed some properties of several Smarandache type funcitons that are involved in many proposed, solved and unsolved problems [2]. An exact formula for counting primes less than or equal to given x in terms of classical Smarandache function has been discovered by L.Seagull[3]. Ruiz and Perez have quoted this result along with a proof while discussing some properties of the classical Smrandache function (Property 2.4) [2]. §2. A formula for twin prime pairs We now proceed to obtain an exact formula for counting twin prime pairs less than or equal to given x in terms of the classical Smarandache function. We denote by T2 (x) the exact number of twin prime pairs less than or equal to x. Also [m] denotes the integral part of m. Theorem. X · S(j) · S(j + 2) ¸ T2 (x) = −1 + , (j) · (j + 2) 1≤j≤x−2 where S(k) denotes the value of classical Smarandache function evaluated at k. Proof. It is well known that (1) S(p) = p iff p is prime > 4, (2) S(p) < p when p is not prime and p 6= 4, (3) S(4) = 4. Vol. 1 The classical Smarandache function and a formula for twin primes 115 In the h light of i the above properties, (1) S(2)·S(4) = 1, (2)·(4) therefore (2, 4) will be counted as a twin prime pair in the sum given in the above formula. The term h ” − 1” is iadded in the formula to eliminate this additional count. Also, (2) S(j)·S(j+2) =1 (j)·(j+2) only when (j, j + 2) will be a twin prime pair and in all other cases h i S(j)·S(j+2) = 0. (j)·(j+2) Hence the theorem is obvious. Let us denote by T2n (x) the exact number of prime pairs (p, p + 2n), n is a positive integer and n > 1. Corollary. T2n (x) = X 1≤j≤x−2n · ¸ S(j) · S(j + 2n) . (j) · (j + 2n) Proof. Since n > 1, the illegal appearance of the pair (2, 4) as a prime pair is automatically prohibited, and the proof follows by proceeding on the similar lines. §3. Conclusion Like formula for counting primes up to given x, [3], one can obtain a similar formula for counting twin prime pairs as well as prime pairs in which the primes are separated by 2n in terms of the Classical Smarandache Functions by proceeding along the same lines. Acknowledgements The author is thankful to Dr. M.R.Modak and Dr. S.A.Katre, Bhaskaracharya Pratishthana, Pune, for their keen interest. References [1] e-books at http://www.gallup.unm.edu/ smarandache/math.htm. [2] Sebastian Martin Ruiz and M.Perez, Properties and Problems Ralated to the Smarandache Type Functions, Mathematics Magazine, 2(2004). [3] L.Seagull, The Smarandache Function and the Number of Primes up to X, Mathematical Spectrum, University of Shielfield, 3(1995), 53. Scientia Magna Vol. 1 (2005), No. 2, 116-119 On completely f -perfect numbers József Sándor Babeş-Bolyai University of Cluj, Romania 1. Let f : N∗ → N be a given arithmetic function. Recently, J.L.Pe[3] has called a number n to be f -perfect, if X f (i) = n, (1) i|n,i<n where the sum is taken for all proper divisors i of n (i.e. i|n, i < n). Clearly for f = I (where I(n) = n for all n ≥ 1)(1) given σ(n) = 2n, i.e. one reobtains the classical perfect numbers. Let S, Z be the Smarandache, resp. Pseudo-Smarandache functions, defined by S(n) = min{k²N : n|n!}, Z(n) = min{k²N : n| k(k + 1) } 2 (2) Since 0! = 1, we may assume S(1) = 0. With this assumption, recently Ch. Ashbacher [1] showed that for n ≤ 106 the only S-perfect number is n = 12, while the Z-perfect numbers in this range are n = 4, 6, 471544. 2. In what follows, we shall call a number n completely f -perfect, if X f (i) = n, (3) i|n where the sum is over all divisors of n. We note that this notion generalizes again the classical notion of a perfect number, since for f = IX − ϕ (where ϕ is Euler’s totient), clearly f (n) = n − ϕ ≥ 0 for all n, and by Gauss’ relation ϕ(i) = n, (3) implies σ(n) = 2n. Thus, i|n the completely I − ϕ-perfect numbers are the perfect numbers. 3. By assuming S(1) = 0, P.Gronas [2] has shown that for f = S, all solutions of equation (3) are the following: n = p (prime), and n = 9, 16, 24. Thus: Theorem 1. All completely S-perfect numbers are the primes, and the numbers 9,16,24. Remark. It is important to note, that if one defines S(n) by S(n) = min{k²N∗ : n|k!}, then clearly S(1) = 1, and Theorem 1 above, as well as Aschbacher’s result, are no more valid. Indeed, when S(1) = 0, then for f = S, (1) has the form X (4) S(i) = n, i|n,1<i<n Vol. 1 On completely f -perfect numbers 117 while if S(1) = 1, then (1) becomes X f (i) = n − 1. (5) i|n,1<i<n Thus we have two distinct equations, namely (4) at one part, and (5) at another part. On the other hand, from (3) we can deduce the two distinct equations (the first one solved by Theorem 1): X f (i) = n − S(n), (6) i|n,1<i<n and X f (i) = n − S(n) − 1. (7) i|n,1<i<n Then, since S(2) = 2, S(3) = 3 and 2, 3 are the only proper divisors of 6, n = 6 is a solution to (5), but not (4). Therefore one can have two distinct notions of ”S-perfect” (as well as ”completely S-perfect”) numbers. Let us call n to be S-perfect in the case 1, if (4) holds, and S-perfect in the sense 2, if (5) holds. The following little result is true: Theorem 2. Let p,q be distinct primes. Then the only S-perfect number n of the form n = pq in the sense 2 is n = 6. There are no S-perfect numbers of this form in sense 1. The only S-perfect numbers of this form in sense 2. Proof. Let n = pq in (5), and assume p < q. Then since S(p) = p, S(q) = q, one obtains the equation p + q = pq − 1 i.e. (p − 1)(q − 1) = 2, giving p − 1 = 1,q − 1 = 2, i.e. p = 2, q = 3, implying n = 6. The equation (4) gives p + q = pq, which cannot have a solution. Let now n = p2 q. The proper divisors are p, q, p2 , pq, and since S(p2 ) = 2p, S(pq) = q, (4) implies the equation 3p + 2q = p2 q. Since p|2q, clearly p|2, so p = 2. This implies q = 3, so n = 22 · 3 = 12. The equation 3p + 2q = p2 q − 1 can not have solution, since for p = 2 this gives 7 = 2q (impossible); while for p, q odd, p2 q −1 = even, 3p + 2q = odd. In the similar way, one can prove: Thoerem 3. There are no completely S-perfect numbers of the form n = pq in both sense. There are no completely S-perfect numbers of the form n = p2 q in sense 1. The only completely S-perfect number of this form in sense 2, is n = 28. Proof. Let n = pq (p > q primes) in (6), resp. (7). Then one gets p + q = pq − q, resp. p + q = pq − q − 1. The first equation, i.e. p + 2q = pq forces q|p, impossible; while the second one, i.e. p + 2q + 1 = pq for p = 2 gives 3 = 0, while for p,q ≥ 3 left side = even, right side n odd. Now let n = p2 q. Since S(p2 ) = 2p, S(pq) = q and S(p2 q) = max{S(p2 ), S(q)} = max{2p, q}, one can deduce the following equations: 118 József Sándor No. 2 i) 3p + 2q = p2 q − max{2p, q}; ii)3p + 2q = p2 q − max{2p, q} − 1. i) a) 2p > q ⇒ 5p + 2q = p2 q. Since p|2q, this gives p = 2, when 2q = 10, impossible. b) 2p < q ⇒ 3p + 3q = p2 q, giving p|3q, so p = 3 and 9 = 6p, impossible again. ii)a) ⇒ 5p + 2q = p2 q − 1. For p = 2 one has q = −5, impossible, while for p, q ≥ 3 left side = odd, right side = even. b) ⇒ 3p + 3q = p2 q − 1. Remark that p = 2,q = 7 is a solution of this equation and this satisfies condition 2p < q since 4 < 7. Now, for p, q ≥ 3 write the equation in the form q(pq − 3) = 3p + 1 and remark that by q > p ≥ 3 one has q ≥ 5 so q(pq − 3) ≥ 5(5p − 3) > 3p + 1, i.e. 22p > 16, which is true. Thus, there are no other solutions. 4. The solutions n = 6 of (5) and n = 28 of (7) are ordinary prefect numbers. Having in view to determine all these solutions, we first prove the following result: Theorem 4. Let n = 2k p, where p is an odd prime, k ≥ 1 and p ≥ 2k. Then n cannot be a solution to equation (4) or (6). The number n is a solution of (5) iff n = 6. The only solution of this type of equation (7) is n = 28. X Proof. We first calculate S = S(n). Since the proper divisors of n = 2k p are i|n,1<i<n 2, 22 , · · · , 2k , p, 2p, 22 p, · · · , 2k−1 p, one has S(n) = S(2) + S(22 ) + · · · + S(2k ) + S(1 · p) + S(2 · p) + · · · + S(2k−1 · p) Now S(2l p) = max{S(2l ), S(p)} = max{S(2l ), p} and since it is well-known that S(2l ) ≤ 2l, by 2l ≤ 2(k − 1) < 2k < p we get S ≤ 2 + 2 · 2 + · · · + 2 · k + kp = 2(k + 1)k + kp = k(k + 1) + kp, 2 so S ≤ k(k + 1) + kp Therefore, by (4), (5), (6), (7) we have to solve the equations   2k p      2k p − 1 S(2) + S(22 ) + · · · + S(2k ) + kp =   2k p − p     k 2 p−p−1 (8) (40 ) (50 ) (60 ) (70 ) a) For (4’) remark that by (8) we must have 2k p ≤ kp + k(k + 1), so p(2k − k) ≤ k(k + 1). Since p > 2k, on the other hand we have p(2k − k) > 2k(2k − 2) ≥ k(k + 1) by the inequality 2(2k − k) ≥ k + 1, i.e. 2k+1 ≥ 3k + 1, k ≥ 1 (9) Vol. 1 On completely f -perfect numbers 119 It is easy to verify by induction that (9) holds true for all k ≥ 1. Therefore, equation (4’) is impossible. Remark. The solution n = 12 = 22 · 3 with p = 3, k = 2 doesn’t satisfy p > 2k. b) Similarly, for (5’), by (8) we should have satisfied the inequality 2k p − 1 ≤ kp + k(k + 1). Now, by p > 2k we get p(2k − k) < 2k(2k − k) > k(k + 1) + 1 ⇔ k(2k+1 − 3k − 1) ≥ 2. Now, the inequality 2k+1 ≥ 3k + 2, k ≥ 2 (10) holds true. Thus for k ≥ 2 we cannot have a solution. For k = 1, however, by Theorem 2 we get the solution n = 21 · 3 when p = 3 > 2 · 1 = 2. c) For (6’) remark, that similarly we must have 2k p − p ≤ k(k + 1) + kp, or p(2k − k − 1) ≤ k(k + 1). Now, by p > 2k, and the inequality 2k+1 > 3k + 3, k ≥ 3 (11) it follows that p(2k − k − 1) > 2k(2k − k − 1) > k(k + 1). Thus we could have eventually k = 1 or k = 2. By Theorem 3 we cannot have solutions. d) The equation (7’), by (8) implies 2k p−p−1 ≤ k(k+1)+kp so p(2k −k−1)−1 ≤ k(k+1). Now, by p > 2k, and 2k(2k − k − 1) > k(k + 1) + 1 ⇔ k(2k+1 − 3k − 3) > 1, this is true by 2k+1 ≥ 3k + 4, k ≥ 3, (12) so we could have eventually k = 1 or k = 2, i.e. n = 2p or n = 22 p. By Theorem 3 this is possible only when p = 7, when p > 2k, i.e. 7 > 4 is satisfied. Corollary. There are no ordinary even perfect numbers which are S-perfect or completely S-perfect in sense 1. The only even perfect number which is S-perfect in sense 2 is n = 6. The only even perfect number which is completely S-perfect in sense 2 is n = 28. Proof. Let n be an even perfect number. Then, by Euclid-Euler’s theorem, n can be written as n = 2k p, where p is a prime of the form p = 2k+1 − 1. Now, p > 2k is true, since 2k+1 > 2k + 1, k ≥ 1. This follows e.g. by induction, and we omit the details. Theorem 4 implies the corollary. 4. Finally, note that in paper [4] we have proved that the only completely d-perfect numbers are n = 1, 3, 18 and 36 (here d(n) is the number of distinct divisors of n). References [1] Ch.Ashbacher, On numbers that are pseudo-Smarandache and Smarandache perfect, Smarandache Notions Journal, 41(2004), 40-42. [2] P.Gronas, The solution of the diophantine equation σ(n) = n, Smarandanche Notions Journal, 4-5(1994), 14-16. [3] J.L.Pe, On a generalization of perfect numbers, J.Recr. Math. (to appear). [4] J.Sándor, On completely d-perfect numbers(to appear). Scientia Magna Vol. 1 (2005), No. 2, 120-133 Parallel bundles in planar map geometries Linfan Mao Institute of Systems, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, P.R.China E-mail: maolinfan@163.com Abstract Parallel lines are very important objects in Euclid plane geometry and its behaviors can be gotten by one’s intuition. But in a planar map geometry, a kind of the Smarandache geometries, the situation is complex since it may contains elliptic or hyperbolic points. This paper concentrates on the behaviors of parallel bundles in planar map geometries, a generalization of parallel lines in plane geometry and obtains characteristics for parallel bundles. Keywords Parallel bundle; Planar map; Smarandache geometry; Map geometry; Classification. §1. Introduction A map is a connected topological graph cellularly embedded in a surface. On the past century, many works are concentrated on to find the combinatorial properties of maps, such as to determine whether exists a particularly embedding on a surface ([7], [11]) or to enumerate a family of maps ([6]). All these works are on the side of algebra, not the object itself, i.e., geometry. For the later, more attentions are given to its element’s behaviors, such as, the line, angle, area, curvature, · · · , see also [12] and [14]. For returning to its original face, the conception of map geometries is introduced in [10]. It is proved in [10] that the map geometries are nice model of the Smarandache geometries. They are also a new kind of intrinsic geometry of surfaces ([1]). The main purpose of this paper is to determine the behaviors of parallel bundles in planar geometries, a generalization of parallel lines in the Euclid plane geometry. An axiom is said Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways. A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969)([5], [13]). In [3] and [4], Iseri presented a nice model of the Smarandache geometries, called smanifolds by using equilateral triangles, which is defined as follows([3], [5] and [9]): An s-manifold is any collection C(T, n) of these equilateral triangular disks Ti , 1 ≤ i ≤ n satisfying the following conditions: (i) Each edge e is the identification of at most two edges ei , ej in two distinct triangular disks Ti , Tj , 1 ≤ i, j ≤ n and i 6= j; (ii) Each vertex v is the identification of one vertex in each of five, six or seven distinct triangular disks. Vol. 1 Parallel bundles in planar map geometries 121 The conception of map geometries without boundary is defined as follows ([10]). Definition 1.1 For a given combinatorial map M , associates a real number µ(u), 0 < µ(u) < π, to each vertex u, u ∈ V (M ). Call (M, µ) a map geometry without boundary, µ(u) the angle factor of the vertex u and to be orientablle or non-orientable if M is orientable or not. In [10], it has proved that map geometries are the Smarandache geometries. The realization of each vertex u, u ∈ V (M ) in R3 space is shown in the Fig.1 for each case of ρ(u)µ(u) > 2π, = 2π or < 2π, call elliptic point, euclidean point and hyperbolic point, respectively. ρ(u)µ(u) < 2π ρ(u)µ(u)¸ = 2π ρ(u)µ(u) > 2π Fig.1 ¸ Therefore, a line passes through an elliptic vertex, an euclidean vertex or a hyperbolic vertex u has angle ρ(u)µ(u) at the vertex u. It is not 180◦ if the vertex u is elliptic or hyperbolic. Then 2 what is the angle of a line passes through a point on an edge of a map? It is 180◦ ? Since we wish the change of angles on an edge is smooth, the answer is not. For the Smarandache geometries, the parallel lines in them are need to be given more attention. We have the following definition. Definition 1.2 A family L of infinite lines not intersecting each other in a planar geometry is called a parallel bundle. In the Fig.2, we present all cases of parallel bundles passing through an edge in planar geometries, where, (a) is the case of points u, v are same type with ρ(u)µ(u) = ρ(v)µ(v), (b) and (c) the cases of same types with ρ(u)µ(u) > ρ(v)µ(v) and (d) the case of u is elliptic and v hyperbolic. Fig.2 ¸ Here, we assume the angle at the intersection point is in clockwise, that is, a line passing 122 Linfan Mao No. 2 through an elliptic point will bend up and a hyperbolic point will bend down, such as the cases − → (b),(c) in the Fig.2. For a vector O on the Euclid plane, call it an orientation. We classify − → parallel bundles in planar map geometries along an orientation O . §2. A condition for parallel bundles We investigate the behaviors of parallel bundles in the planar map geometries. For this object, we define a function f (x) of angles on an edge of a planar map as follows. Definition 2.1 Denote by f (x) the angle function of a line L passing through an edge uv at the point of distance x to u on the edge uv. Then we get the following result. Proposition 2.1 A family L of parallel lines passing through an edge uv is a parallel bundle iff ¯ df ¯¯ ≥ 0. dx ¯+ Proof. If L is a parallel bundle, then any two lines L1 , L2 will not intersect after them passing through the edge uv. Therefore, if θ1 , θ2 are the angles of L1 , L2 at the intersect points of L1 , L2 with uv and L2 is far from u than L2 , then we know that θ2 ≥ θ1 . Whence, for any point with x distance from u and ∆x > 0, we have that f (x + ∆x) − f (x) ≥ 0. Therefore, we get that ¯ df ¯¯ f (x + ∆x) − f (x) ≥ 0. = lim ¯ dx + ∆x→+0 ∆x As the cases in the Fig.1. ¯ df ¯ Now if dx ¯ ≥ 0, then f (y) ≥ f (x) if y ≥ x. Since L is a family of parallel lines before + meeting uv, whence, any two lines in L will not intersect each other after them passing through uv. Therefore, L is a parallel bundle. \ A general condition for a family of parallel lines passing through a cut of a planar map being a parallel bundle is the following. Proposition 2.2 Let (M, µ) be a planar map geometry, C = {u1 v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2 v2 , · · · , ul vl from the left to the right, l ≥ 1 and the angle functions on them are f1 , f2 , · · · , fl , respectively, also see the Fig.3. Vol. 1 Parallel bundles in planar map geometries 123 Fig.3 ¸ Then a family L of parallel lines passing through C is a parallel bundle iff for any x, x ≥ 0, f10 (x) ≥ 0 0 0 f1+ (x) + f2+ (x) ≥ 0 0 0 0 f1+ (x) + f2+ (x) + f3+ (x) ≥ 0 ············ 0 f1+ (x) + 0 f2+ (x) 0 + · · · + fl+ (x) ≥ 0. Proof. According to the Proposition 2.1, see the following Fig.4, Fig.4 ¸ we know that any lines will not intersect after them passing through u1 v1 and u2 v2 iff for ∀∆x > 0 and x ≥ 0, 0 f2 (x + ∆x) + f1+ (x)∆x ≥ f2 (x). That is, 0 0 f1+ (x) + f2+ (x) ≥ 0. Similarly, any lines will not intersect after them passing through u1 v1 , u2 v2 and u3 v3 iff for ∀∆x > 0 and x ≥ 0, 0 0 f3 (x + ∆x) + f2+ (x)∆x + f1+ (x)∆x ≥ f3 (x). 124 Linfan Mao No. 2 That is, 0 0 0 f1+ (x) + f2+ (x) + f3+ (x) ≥ 0. Generally, any lines will not intersect after them passing through u1 v1 , u2 v2 , · · · , ul−1 vl−1 and ul vl iff for ∀∆x > 0 and x ≥ 0, 0 0 fl (x + ∆x) + fl−1+ (x)∆x + · · · + f1+ (x)∆x ≥ fl (x). Whence, we get that 0 0 0 f1+ (x) + f2+ (x) + · · · + fl+ (x) ≥ 0. Therefore, a family L of parallel lines passing through C is a parallel bundle iff for any x, x ≥ 0, we have that f10 (x) ≥ 0 0 0 f1+ (x) + f2+ (x) ≥ 0 0 0 0 f1+ (x) + f2+ (x) + f3+ (x) ≥ 0 ············ 0 f1+ (x) + 0 f2+ (x) 0 + · · · + fl+ (x) ≥ 0. This completes the proof. \. Corollary 2.1 Let (M, µ) be a planar map geometry, C = {u1 v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2 v2 , · · · , ul vl from the left to the right, l ≥ 1 and the angle functions on them are f1 , f2 , · · · , fl . Then a family L of parallel lines passing through C is still parallel lines after them leaving C iff for any x, x ≥ 0, f10 (x) ≥ 0 0 0 f1+ (x) + f2+ (x) ≥ 0 0 0 0 f1+ (x) + f2+ (x) + f3+ (x) ≥ 0 ············ 0 f1+ (x) + 0 f2+ (x) 0 + · · · + fl−1+ (x) ≥ 0 0 0 0 f1+ (x) + f2+ (x) + · · · + fl+ (x) = 0. Proof. According to the Proposition 2.2, we know the condition is a necessary and sufficient condition for L being a parallel bundle. Now since lines in L are parallel lines after them leaving C iff for any x ≥ 0 and ∆x ≥ 0, there must be that 0 0 fl (x + ∆x) + fl−1+ (x)∆x + · · · + f1+ (x)∆x = fl (x). Therefore, we get that Vol. 1 125 Parallel bundles in planar map geometries 0 0 0 f1+ (x) + f2+ (x) + · · · + fl+ (x) = 0 \ When do the parallel lines parallel the initial parallel lines after them passing through a cut C in a planar map geometry? The answer is in the following result. Proposition 2.3 Let (M, µ) be a planar map geometry, C = {u1 v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2 v2 , · · · , ul vl from the left to the right, l ≥ 1 and the angle functions on them are f1 , f2 , · · · , fl . Then the parallel lines parallel the initial parallel lines after them passing through C iff for ∀x ≥ 0, f10 (x) ≥ 0 0 0 f1+ (x) + f2+ (x) ≥ 0 0 0 0 f1+ (x) + f2+ (x) + f3+ (x) ≥ 0 ············ 0 f1+ (x) + 0 f2+ (x) 0 + · · · + fl−1+ (x) ≥ 0 and f1 (x) + f2 (x) + · · · + fl (x) = lπ. Proof. According to the Proposition 2.2 and Corollary 2.1, we know the parallel lines passing through C is a parallel bundle. We calculate the angle α(i, x) of a line L passing through an edge ui vi , 1 ≤ i ≤ l with the line before it meeting C at the intersection of L with the edge ui vi , where x is the distance of the intersection point to u1 on u1 v1 , see also the Fig.4. By the definition, we know the angle α(1, x) = f (x) and α(2, x) = f2 (x) − (π − f1 (x)) = f1 (x) + f2 (x) − π. Now if α(i, x) = f1 (x) + f2 (x) + · · · + fi (x) − (i − 1)π, then similar to the case i = 2, we know that α(i + 1, x) = fi+1 (x) − (π − α(i, x)) = fi+1 (x) + α(i, x) − π. Whence, we get that α(i + 1, x) = f1 (x) + f2 (x) + · · · + fi+1 (x) − iπ. Notice that a line L parallel the initial parallel line after it passing through C iff α(l, x) = π, i.e., f1 (x) + f2 (x) + · · · + fl (x) = lπ. This completes the proof. \ §3. Linear condition and combinatorial realization for parallel bundles For the simplicity, we can assume the function f (x) is linear and denoted it by fl (x). We can calculate fl (x) as follows. 126 Linfan Mao No. 2 Proposition 3.1 The angle function fl (x) of a line L passing through an edge uv at the point with distance x to u is fl (x) = (1 − x ρ(u)µ(v) x ρ(v)µ(v) ) + , d(uv) 2 d(uv) 2 where, d(uv) is the length of the edge uv. Proof. Since fl (x) is linear, we know that fl (x) satisfies the following equation. ρ(u)µ(u) 2 ρ(u)µ(u) ρ(v)µ(v) − 2 2 fl (x) − = x , d(uv) Calculation shows that fl (x) = (1 − ρ(u)µ(v) x ρ(v)µ(v) x ) + . \ d(uv) 2 d(uv) 2 Corollary 3.1 Under the linear assumption, a family L of parallel lines passing through an edge uv is a parallel bundle iff µ(v) ρ(u) ≤ . ρ(v) µ(u) Proof. According to the Proposition 2.1, a family of parallel lines passing through an edge uv is a parallel bundle iff for ∀x, x ≥ 0, f 0 (x) ≥ 0, i.e., ρ(v)µ(v) ρ(u)µ(u) − ≥ 0. 2d(uv) 2d(uv) Therefore, a family L of parallel lines passing through an edge uv is a parallel bundle iff ρ(v)µ(v) ≥ ρ(u)µ(u). Whence, ρ(u) µ(v) ≤ . ρ(v) µ(u) \ For a family of parallel lines pass through a cut, we have the following condition for it being a parallel bundle. Proposition 3.2 Let (M, µ) be a planar map geometry, C = {u1 v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2 v2 , · · · , ul vl from the left to the right, l ≥ 1. Then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle iff the angle factor µ satisfies the following linear inequality system ρ(v1 )µ(v1 ) ≥ ρ(u1 )µ(u1 ) ρ(v1 )µ(v1 ) ρ(v2 )µ(v2 ) ρ(u1 )µ(u1 ) ρ(u2 )µ(u2 ) + ≥ + d(u1 v1 ) d(u2 v2 ) d(u1 v1 ) d(u2 v2 ) ············ Vol. 1 Parallel bundles in planar map geometries ρ(v1 )µ(v1 ) d(u1 v1 ) + ≥ 127 ρ(v2 )µ(v2 ) ρ(vl )µ(vl ) + ··· + d(u2 v2 ) d(ul vl ) ρ(u1 )µ(u1 ) ρ(u2 )µ(u2 ) ρ(ul )µ(ul ) + + ··· + . d(u1 , v1 ) d(u2 , v2 ) d(ul , vl ) Proof. Under the linear assumption, for any integer i, i ≥ 1, we know that 0 fi+ (x) = ρ(vi )µ(vi ) − ρ(ui )µ(ui ) 2d(ui vi ) by the Proposition 3.1. Whence, according to the Proposition 2.2, we get that a family L of parallel lines passing through C is a parallel bundle iff the angle factor µ satisfies the following linear inequality system ρ(v1 )µ(v1 ) ≥ ρ(u1 )µ(u1 ) ρ(v1 )µ(v1 ) ρ(v2 )µ(v2 ) ρ(u1 )µ(u1 ) ρ(u2 )µ(u2 ) + ≥ + d(u1 v1 ) d(u2 v2 ) d(u1 v1 ) d(u2 v2 ) ············ ρ(v1 )µ(v1 ) d(u1 v1 ) + ≥ ρ(v2 )µ(v2 ) ρ(vl )µ(vl ) + ··· + d(u2 v2 ) d(ul vl ) ρ(u1 )µ(u1 ) ρ(u2 )µ(u2 ) ρ(ul )µ(ul ) + + ··· + . d(u1 , v1 ) d(u2 , v2 ) d(ul , vl ) This completes the proof. \ For planar maps underlying a regular graph, we have the following interesting results for parallel bundles. Corollary 3.2 Let (M, µ) be a planar map geometry with M underlying a regular graph, C = {u1 v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2 v2 , · · · , ul vl from the left to the right, l ≥ 1. Then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle iff the angle factor µ satisfies the following linear inequality system µ(v1 ) ≥ µ(u1 ) µ(v1 ) µ(v2 ) µ(u1 ) µ(u2 ) + ≥ + d(u1 v1 ) d(u2 v2 ) d(u1 v1 ) d(u2 v2 ) ············ µ(v1 ) µ(v2 ) µ(vl ) µ(u1 ) µ(u2 ) µ(ul ) + + ··· + ≥ + + ··· + d(u1 v1 ) d(u2 v2 ) d(ul vl ) d(u1 v1 ) d(u2 v2 ) d(ul vl ) and particularly, if assume that all the lengths of edges in C are the same, then 128 Linfan Mao No. 2 µ(v1 ) ≥ µ(u1 ) µ(v1 ) + µ(v2 ) ≥ µ(u1 ) + µ(u2 ) ······ ··· ······ µ(v1 ) + µ(v2 ) + · · · + µ(vl ) ≥ µ(u1 ) + µ(u2 ) + · · · + µ(ul ). Certainly, by choosing different angle factors, we can also get combinatorial conditions for existing parallel bundles under the linear assumption. Proposition 3.3 Let (M, µ) be a planar map geometry, C = {u1 v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2 v2 , · · · , ul vl from the left to the right, l ≥ 1. If for any integer i, i ≥ 1, µ(vi ) ρ(ui ) ≤ , ρ(vi ) µ(ui ) then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle. Proof. Notice that under the linear assumption, for any integer i, i ≥ 1, we know that 0 fi+ (x) = ρ(vi )µ(vi ) − ρ(ui )µ(ui ) 2d(ui vi ) 0 by the Proposition 3.1. Whence, fi+ (x) ≥ 0 for i = 1, 2, · · · , l. Therefore, we get that f10 (x) ≥ 0 0 0 f1+ (x) + f2+ (x) ≥ 0 0 0 0 f1+ (x) + f2+ (x) + f3+ (x) ≥ 0 ············ 0 f1+ (x) + 0 f2+ (x) 0 + · · · + fl+ (x) ≥ 0. By the Proposition 2.2, we know that a family L of parallel lines passing through C is a parallel bundle. \ §4. Classification of parallel bundles For a cut C in a planar map geometry and e ∈ C, denote by fe (x) the angle function on P the edge e, f (C, x) = fe (x). If f (C, x) is independent on x, then we abbreviate it to f (C). e∈C According to the results in the Section 2 and 3, we can classify the parallel bundles with a − → given orientation O in planar map geometries into the following 15 classes, where, each class is labelled by a 4-tuple 0, 1 code. Classification of parallel bundles − → (1) C1000 : for any cut C along O , f (C) = |C|π; Vol. 1 Parallel bundles in planar map geometries 129 − → (2) C0100 : for any cut C along O , f (C) < |C|π; − → (3) C0010 : for any cut C along O , f (C) > |C|π ; − → 0 (4) C0001 : for any cut C along O , f+ (C, x) > 0 for ∀x, x ≥ 0; − → (5) C1100 : There exist cuts C1 , C2 along O , such that f (C1 ) = |C1 |π and f (C2 ) = c < |C2 |π; − → (6) C1010 : there exist cuts C1 , C2 along O , such that f (C1 ) = |C1 |π and f (C2 ) > |C2 |π; − → 0 (7) C1001 : there exist cuts C1 , C2 along O , such that f (C1 ) = |C1 |π and f+ (C2 , x) > 0 for ∀x, x ≥ 0; − → (8) C0110 : there exist cuts C1 , C2 along O , such that f (C1 ) < |C1 |π and f (C2 ) > |C2 |π; − → 0 (9) C0101 : there exist cuts C1 , C2 along O , such that f (C1 ) < |C1 |π and f+ (C2 , x) > 0 for ∀x, x ≥ 0; − → 0 (10) C0011 : there exist cuts C1 , C2 along O , such that f (C1 ) > |C1 |π and f+ (C2 , x) > 0 for ∀x, x ≥ 0; − → (11) C1110 : there exist cuts C1 , C2 and C3 along O , such that f (C1 ) = |C1 |π, f (C2 ) < |C2 |π and f (C3 ) > |C3 |π; − → (12) C1101 : there exist cuts C1 , C2 and C3 along O , such that f (C1 ) = |C1 |π, f (C2 ) < 0 |C2 |π and f+ (C3 , x) > 0 for ∀x, x ≥ 0; − → (13) C1011 : there exist cuts C1 , C2 and C3 along O , such that f (C1 ) = |C1 |π, f (C2 ) > 0 |C2 |π and f+ (C1 , x) > 0 for ∀x, x ≥ 0; − → (14) C0111 : there exist cuts C1 , C2 and C3 along O , such that f (C1 ) < |C1 |π, f (C2 ) > 0 |C2 |π and f+ (C1 , x) > 0 for ∀x, x ≥ 0; − → (15) C1111 : there exist cuts C1 , C2 , C3 and C4 along O , such that f (C1 ) = |C1 |π, f (C2 ) < 0 (C4 , x) > 0 for ∀x, x ≥ 0. |C2 |π, f (C3 ) > |C3 |π and f+ Notice that only the first three classes may be parallel lines after them passing through the cut C. All of the other classes are only parallel bundles, not parallel lines in the usual meaning. − → Proposition 4.1 For an orientation O , the 15 classes C1000 ∼ C1111 are all the parallel bundles in planar map geometries. Proof. Not loss of generality, we assume C1 , C2 , · · · , Cm , m ≥ 1, are all the cuts along − → − → O in a planar map geometry (M, µ) from the upon side of O to its down side. We find their structural characters for each case in the following discussion. C1000 : By the Proposition 2.3, a family L of parallel lines parallel their initial lines before meeting M after the passing through M . − → C0100 : By the definition, a family L of parallel lines is a parallel bundle along O only if f (C1 ) ≤ f (C2 ) ≤ · · · ≤ f (Cm ) < π. Otherwise, some lines in L will intersect. According to the Corollary 2.1, they parallel each other after they passing through M only if f (C1 ) = f (C2 ) = · · · = f (Cm ) < π. − → C0010 : Similar to the case C0100 , a family L of parallel lines is a parallel bundle along O only if 130 Linfan Mao No. 2 π < f (C1 ) ≤ f (C2 ) ≤ · · · ≤ f (Cm ) and parallel each other after they passing through M only if π < f (C1 ) = f (C2 ) = · · · = f (Cm ). C0001 : Notice that by the proof of the Proposition 2.3, a line has angle f (C, x) − (|C| − 1)π after it passing through C with the initial line before meeting C. In this case, a family L of − → parallel lines is a parallel bundle along O only if for ∀xi , xi ≥ 0, 1 ≤ i ≤ m, f (C1 , x1 ) ≤ f (C2 , x2 ) ≤ · · · ≤ f (Cm , xm ). Otherwise, they will intersect. − → C1100 : In this case, a family L of parallel lines is a parallel bundle along O only if there is an integer k, 2 ≤ k ≤ m, such that f (C1 ) ≤ f (C2 ) ≤ · · · ≤ f (Ck−1 ) < f (Ck ) = f (Ck+1 ) = · · · = f (Cm ) = π. Otherwise, they will intersect. C1010 : Similar to the case C1100 , in this case, a family L of parallel lines is a parallel bundle − → along O only if there is an integer k, 2 ≤ k ≤ m, such that π = f (C1 ) = f (C2 ) = · · · = f (Ck ) < f (Ck+1 ) ≤ · · · ≤ f (Cm ). Otherwise, they will intersect. − → C1001 : In this case, a family L of parallel lines is a parallel bundle along O only if there is an integer k, l, 1 ≤ k < l ≤ m, such that for ∀xi , xi ≥ 0, 1 ≤ i ≤ k or l ≤ i ≤ m, f (C1 , x1 ) ≤ f (C2 , x2 ) ≤ · · · ≤ f (Ck , xk ) < f (Ck+1 ) = f (Ck+2 ) = · · · = f (Cl−1 ) = π < f (Cl , xl ) ≤ · · · ≤ f (Cm , xm ). Otherwise, they will intersect. − → C0110 : In this case, a family L of parallel lines is a parallel bundle along O only if there is integers k, 1 ≤ k < m, such that f (C1 ) ≤ f (C2 ) ≤ · · · ≤ f (Ck ) < π < f (Ck+1 ) ≤ · · · ≤ f (Cm ). Otherwise, they will intersect. − → C0101 : In this case, a family L of parallel lines is a parallel bundle along O only if there is integers k, 1 ≤ k ≤ m, such that for ∀xi , xi ≥ 0, 1 ≤ i ≤ m, f (C1 , x1 ) ≤ f (C2 , x2 ) ≤ · · · ≤ f (Ck , xk ) < π ≤ f (Ck+1 , xk+1 ) ≤ · · · ≤ f (Cm , xm ), and there must be a constant in f (C1 , x1 ), f (C2 , x2 ), · · · , f (Ck , xk ). Vol. 1 Parallel bundles in planar map geometries 131 C0011 : In this case, the situation is similar to the case C0101 and there must be a constant in f (Ck+1 , xk+1 ), f (Ck+2 , xk+2 ), · · · , f (Cm , xm ). − → C1110 : In this case, a family L of parallel lines is a parallel bundle along O only if there is an integer k, l, 1 ≤ k < l ≤ m, such that f (C1 ) ≤ f (C2 ) ≤ · · · ≤ f (Ck ) < f (Ck+1 ) = · · · = f (Cl−1 ) = π < f (Cl ) ≤ · · · ≤ f (Cm ). Otherwise, they will intersect. − → C1101 : In this case, a family L of parallel lines is a parallel bundle along O only if there is an integer k, l, 1 ≤ k < l ≤ m, such that for ∀xi , xi ≥ 0, 1 ≤ i ≤ k or l ≤ i ≤ m, f (C1 , x1 ) ≤ f (C2 , x2 ) ≤ · · · ≤ f (Ck , xk ) < f (Ck+1 ) = · · · = f (Cl−1 ) = π < f (Cl , xl ) ≤ · · · ≤ f (Cm , xm ) and there must be a constant in f (C1 , x1 ), f (C2 , x2 ), · · · , f (Ck , xk ). Otherwise, they will intersect. − → C1011 : In this case, a family L of parallel lines is a parallel bundle along O only if there is an integer k, l, 1 ≤ k < l ≤ m, such that for ∀xi , xi ≥ 0, 1 ≤ i ≤ k or l ≤ i ≤ m, f (C1 , x1 ) ≤ f (C2 , x2 ) ≤ · · · ≤ f (Ck , xk ) < f (Ck+1 ) = · · · = f (Cl−1 ) = π < f (Cl , xl ) ≤ · · · ≤ f (Cm , xm ) and there must be a constant in f (Cl , xl ), f (Cl+1 , xl+1 ), · · · , f (Cm , xm ). Otherwise, they will intersect. − → C0111 : In this case, a family L of parallel lines is a parallel bundle along O only if there is an integer k, 1 ≤ k ≤ m, such that for ∀xi , xi ≥ 0, f (C1 , x1 ) ≤ f (C2 , x2 ) ≤ · · · ≤ f (Ck , xk ) < π < f (Cl , xl ) ≤ · · · ≤ f (Cm , xm ) and there must be a constant in f (C1 , x1 ), f (C2 , x2 ), · · · , f (Ck , xk ) and a constant in f (Cl , xl ), f (Cl+1 , xl+1 ), · · · , f (Cm , xm ). Otherwise, they will intersect. − → C1111 : In this case, a family L of parallel lines is a parallel bundle along O only if there is an integer k, l, 1 ≤ k < l ≤ m, such that for ∀xi , xi ≥ 0, 1 ≤ i ≤ k or l ≤ i ≤ m, f (C1 , x1 ) ≤ f (C2 , x2 ) ≤ · · · ≤ f (Ck , xk ) < f (Ck+1 ) = · · · = f (Cl−1 ) = π < f (Cl , xl ) ≤ · · · ≤ f (Cm , xm ) and there must be a constant in f (C1 , x1 ), f (C2 , x2 ), · · · , f (Ck , xk ) and a constant in f (Cl , xl ), f (Cl+1 , xl+1 ), · · · , f (Cm , xm ). Otherwise, they will intersect. 132 Linfan Mao No. 2 Following the structural characters of the classes C1000 ∼ C1111 , by the Proposition 2.2, 2.3 and Proposition 3.1, we know that any parallel bundle is in one of the classes C1000 ∼ C1111 and each class in C1000 ∼ C1111 is non-empty. This completes the proof. \ A example of parallel bundle in a planar map geometry is shown in the Fig.5, in where the number on a vertex u denotes the number ρ(u)µ(u). Fig.5 ¸ §5. Generalization All the planar map geometries considered in this paper are without boundary. For planar map geometries with boundary, i.e., some faces are deleted ([10]), which are correspondence with the maps with boundary ([2]). We know that they are the Smarandache non-geometries, satisfying one or more of the following conditions: (A1− )It is not always possible to draw a line from an arbitrary point to another arbitrary point. (A2− )It is not always possible to extend by continuity a finite line to an infinite line. (A3− )It is not always possible to draw a circle from an arbitrary point and of an arbitrary interval. (A4− )Not all the right angles are congruent. (A5− )If a line, cutting two other lines, forms the interior angles of the same side of it strictly less than two right angle, then not always the two lines extended towards infinite cut each other in the side where the angles are strictly less than two right angle. Notice that for an one face planar map geometry (M, µ)−1 with boundary, if we choose all points being euclidean, then (M, µ)−1 is just the Poincaré’s model for the hyperbolic geometry. Vol. 1 Parallel bundles in planar map geometries 133 Using the neutrosophic logic idea, we can also define the conception of neutrosophic surface as follow, comparing also with the surfaces in [8] and [14]. Definition 5.1 A neutrosophic surface is a Hausdorff, connected, topological space S such that every point v is elleptic, euclidean, or hyperbolic. For this kind of surface, we present the following problem for the further researching. Problem 5.1 To determine the behaviors of elements, such as, the line, angle, area, · · · , in neutrosophic surfaces. Notice that results in this paper are just the behaviors of line bundles in a neutrosophic plane. References [1]A.D.Aleksandrov and V.A.Zalgaller, Intrinsic geometry of surfaces, American Mathematical Society, 1967. [2]R.P.Bryant and D.Singerman, Foundations of the theory of maps on surfaces with boundary, Quart.J.Math.Oxford, 36(2)(1985), 17-41. [3]H.Iseri, Smarandache manifolds, American Research Press, Rehoboth, NM, 2002. [4]H.Iseri, Partially Paradoxist Smarandache Geometries, http://www.gallup.unm. edu/s̃marandache/Howard-Iseri-paper.htm. [5]L.Kuciuk and M.Antholy, An Introduction to Smarandache Geometries, Mathematics Magazine, Aurora, Canada,12(2003) [6]Y.P.Liu, Enumerative Theory of Maps, Kluwer Academic Publisher, Dordrecht, Boston, London, (1999). [7]Y.P.Liu, Embeddability in Graphs, Kluwer Academic Publisher, Dordrecht, Boston, London, (1995). [8]Mantredo P.de Carmao, Differential Geometry of Curves and Surfaces, Pearson Education asia Ltd, (2004). [9]L.F.Mao, Automorphism groups of maps, surfaces and Smarandache geometries, American Research Press, Rehoboth, NM, 2005. [10]L.F.Mao, A new view of combinatorial maps by Smarandache’s notion, arXiv, Math.GM/0506232, will also appear in Smarandache Notions Journal. [11]B.Mohar and C.Thomassen, Graphs on Surfaces, The Johns Hopkins University Press, London, 2001. [12]V.V.Nikulin and I.R.Shafarevlch, Geometries and Groups, Springer-Verlag Berlin Heidelberg, (1987). [13]F. Smarandache, Mixed noneuclidean geometries, eprint arXiv, math/0010119, 10/2000. [14]J.Stillwell, Classical topology and combinatorial group theory, Springer-Verlag New York Inc., (1980). Scientia Magna Vol. 1 (2005), No. 2, 134-138 On the asymptotic properties of odd sieve sequence Yang Hai† and Fu Ruiqin‡ † Research Center for Basic Science, Xi’an Jiaotong University Xi’an, Shaanxi, P.R.China ‡ School of Science, Department of Mathematics, xi’an Shiyou University Xi’an, Shaanxi, P.R.China Abstract The main purpose of this paper is using elementary method and an analytic method to study the asymptotic properties of the odd sieve sequence, and give two interesting asymptotic formulae. Keywords Odd sieve sequence; Mean value; Asymptotic formula. §1. Introduction And Results According to reference [1], the definition of the odd sieve is: subtract 2 from all prime numbers and obtain a temporary sequence, and choose all odd number that do not belong to the temporary one. For example: 7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 79, 83, 85, 91, 93, 97,. . . are all belong to odd sieve sequence. Let A denote the set of all odd sieve. In reference [1], Professor F.Smarandache asked us to study the properties of the odd sieve sequence. About this problem, it seems that none had studied it, at least we have not seen related paper before. In this paper, we use elementary method and analytic method to study the asymptotic properties of odd sieve sequence, and obtain two interesting asymptotic formulae. That is, we shall prove the following: Theorem 1. For any real number x ≥ 3, we have the asymptotic formula X a = x2 − a∈A a≤x x2 x2 − + O (x) . 2 log x 4 log2 x Theorem 2. For any real number x ≥ 3, we have the asymptotic formula ´ ³ 1 X 1 d(a) = x log x + Bx + O x 2 log2 x , 2 a∈A a≤x where B = 4C − 12 + 32 log 2 − 315ζ(3) 6π 4 , C is Euler constant, d(n) be the Dirichlet divisor function. §2. Several Lemmas To complete the proof of the theorems, we need the following several simple lemmas. Firstly, we have Vol. 1 On the asymptotic properties of odd sieve sequence 135 Lemma 1. For any real number x ≥ 1, we have X nα = n≤x xα+1 + O (xα ) . α+1 (α ≥ 1) Proof. (See reference [2]). Taking α = 0, 1 in the lemma 1, we may immediately obtain the following: Corollary. For any real number x ≥ 1, then we have X 1 = x + O (1) n≤x and X n= n≤x 1 2 x + O (x) . 2 Lemma 2. For any fixed real number x, let π(x) denote the number of all primes not exceeding x, then we have µ ¶ x x x + +O . π(x) = log x log2 x log3 x Proof. (See reference [3]). Lemma 3. For any real number x ≥ 3, let p be a prime, then we have µ 2 ¶ X x2 x2 x p= + +O . 2 log x 4 log2 x log3 x p≤x Proof. From Lemma 2 and by Abel’s identity, we can easily deduce that Z x X p = π(x)x − π(x)dt p≤x 1 µ 2 ¶ x2 x2 x + + O log x log2 x log3 x µZ x ¶ Z x Z x t t t − dt − dt + O dt 2 3 2 log t 2 log t 2 log t µ ¶ x2 x2 x2 = + +O . 2 2 log x 4 log x log3 x = This complets the proof of Lemma 3. Lemma 4. For any real number x ≥ 1, we have X ¡√ ¢ d(n) = x log x + (2C − 1)x + O x , n≤x where C is Euler constant, and d(n) be the Dirichlet divisor function. Proof. This result may be immediately got from [2]. Lemma 5. For any real number x ≥ 1, we have ´ ³ 1 X log 2 3 3 − )x + O x 2 log2 x . d(2n) = x log x + ( 2 2 2 n≤x 136 Yang Hai and Fu Ruiqin No. 2 Proof. Let s = σ + it be a complex number and h(s) = ∞ P n=1 d(2n) ns . Note that d(2n) ¿ n² , so it is clear that h(s) is a Dirichlet series absolutely convergent for Re(s)> 1, by the Euler Product formula [2] and the definition of d(n) we get ∞ YX d(2pm ) pms p m=0 h(s) = ∞ ∞ X d(2m+1 ) Y X d(2pm ) = · 2ms pms m=0 p>2 m=0 ∞ Q P ( = 2ζ 2 (s) · p>2 m=0 d(pm ) pms ) ∞ Q P p m=0 ∞ P = 2ζ 2 (s) · m=0 ∞ P = ζ (s)(2 − ∞ P m=0 d(2m+1 ) 2ms ) d(pm ) pms m+1 d(2 ) 2ms m=0 2 ·( d(2m ) 2ms 1 ), 2s where ζ(s) is the Riemann Zeta-function and Q (1) denotes the product over all primes. p From (1) and Perron’s formula [4], for b = 1 + ², T ≥ 1 and x ≥ 1 we have X n≤x Taking a = 1 2 X n≤x 1 d(2n) = 2πi Z b+iT b−iT ¯ b¯ µ ¶ ¯x ¯ xH(2x) log x xs ¯ ¯ . h(s) ds + O ¯ ¯ + O s T T + ², we move the integral line in (2). Then d(2n) = Res ζ 2 (s)(2 − s=1 1 xs ) 2s s ¯Z Z a+iT Z b+iT ¯¯ 1 ¯¯ a−iT 1 xs ¯ 2 + + + ¯ ¯ ζ (s)(2 − s ) ds 2πi ¯ b−iT 2 s a−iT a+iT ¯ ¯ ¯ b¯ ¯ ¯ xH(2x) log x ¯ ¯x ¯ ¯, + O ¯¯ ¯¯ + O ¯¯ ¯ T T where ¯ ¯Z ¯ a−iT Z b+iT ¯ 1 xs x ¯ 2 ¯ + ¯ ζ (s)(2 − s ) ds ¿ ¯ ¯ b−iT 2 s T a+iT ¯ Z a+iT 1 1 xs ζ 2 (s)(2 − s ) ds ¿ x 2 log2 T. 2 s a−iT (2) Vol. 1 On the asymptotic properties of odd sieve sequence 137 Hence, we have X d(2n) n≤x ¯x¯ 1 xs ¯ ¯ ) + O ¯ ¯ s=1 2s s T ¯ ¯ ¯ ¯ ´ ³ 1 ¯ xb ¯ ¯ log x ¯¯ 2 ¯ ¯ ¯ 2 + O x log T + O ¯ ¯ + O ¯xH(2x) T T ¯ ¯ ¯ s 1 x ¯x¯ = Res ζ 2 (s)(2 − s ) + O ¯ ¯ s=1 2 s¯ T ¯ ³ 1 ´ ¯ log x ¯¯ + O x 2 log2 T + O ¯¯x1+ε . T ¯ Res ζ 2 (s)(2 − = (3) 1 Taking T = x 2 +² in (3), then X ³ 1 ´ ³ 1 ´ 1 xs 2 2 −ε 2 log x ) + O x + O x s=1 2s s ³ 1 ´ 1 xs = Res ζ 2 (s)(2 − s ) + O x 2 log2 x . s=1 2 s d(2n) = n≤x Res ζ 2 (s)(2 − Now we can easily get the residue of the function ζ 2 (s)(2 − s=1 Res ζ 2 (s)(2 − s=1 1 2s ) · xs s at second order pole point 3 log 2 3 1 xs ) = x log x + ( − )x. 2s s 2 2 2 Combining (4) and (5), we immediately get X d(2n) = n≤x ³ 1 ´ log 2 3 3 x log x + ( − )x + O x 2 log2 x . 2 2 2 This completes the proof of Lemma 5. Lemma 6. For any x > 0 , let a be any fixed positive integer, then we have X d(p − a) = 0<p−a≤x ³ ´ 315ζ(3) Y (p − 1)2 −1+ε x + O x(log x) , 2π 4 p2 − p + 1 p|a where ² is any positive integer. Proof. This result may be immediately got from [5]. §3. Proof of the Theorem In this section, we will complete the proof of Theorem. First, we have X a∈A a≤x a = X (2n − 1) − n≤x = 2 X n≤x X (p − 2) p≤x n− X n≤x 1− X p≤x (4) p + 2π(x). (5) 138 Yang Hai and Fu Ruiqin No. 2 From Lemma 1, Lemma 2 and Lemma 3, we have ¶ µ 2 X x a = 2 + O (x) − (x + O (1)) 2 a∈A a≤x µ 2 ¶ x x2 x2 − ( + +O ) 2 log x 4log2 x log3 x µ ¶ x x x + 2( +O ) + log x log2 x log3 x = x2 − x2 x2 − + O (x) . 2 log x 4 log2 x This completes the proof of Theorem 1. Now we will give the proof of Theorem 2. From Lemma 4, Lemma 5 and Lemma 6, we can easily obtain X d(a) = a∈A a≤x X d(2n − 1) − n≤x = X n≤2x X d(p − 2) p≤x d(n) − X d(2n) − n≤x X d(p − 2) p≤x ¡√ ¢ 2x log x + 2(2C − 1 + log 2)x + O x ³ 1 ´ 3 log 2 3 − x log x + ( − )x + O x 2 log2 x 2 2 2 ³ ´ 315ζ(3) −1+ε x + O x(log x) − 6π 4 ³ 1 ´ 1 = x log x + Bx + O x 2 log2 x , 2 = where B = 4C − 21 + 32 log 2 − 315ζ(3) 6π 4 , C is Euler constant. This completes the proof of Theorem 2. References [1] F. Smarandache, Only problems, Not solutions, Chicago, Xiquan Publ. House, 1993. [2] Tom M.Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag, 1976. [3] M.Ram Murty, Problems in Analytic Number Theory, New York, 2001, 35-36. [4] Pan Chengdong and Pan Chengbiao, Foundation of Analytic number Theory, Beijing, Science Press, 1997. [5] B.M.Bredihin, Binary Additive Problems of Indeterminate Type I(Russian), Izv. Akad. Nauk SSSR Ser. Mat., 27(1963), 439-462. Scientia Magna Vol. 1 (2005), No. 2, 139-144 Smarandache quasigroup rings Arun S. Muktibodh Mohota Science College Nagpur, India Abstract In this paper, we have introduced Smarandache quasigroups which are Smarandache nonassociative structures. W.B.Kandasamy [2] has studied groupoid ring and loop ring. We have defined Smarandache quasigroup rings which are again non-associative structures having two binary operations. Substructures of quasigroup rings are also studied. Keywords Non-associative rings; Smarandache non-associative rings; Quasigroups; Smarandache quasigroups; Smarandache quasigroup rings. §1. Introduction In the paper [2] W.B.Kandasamy has introduced a new concept of groupoid rings. This structure provides number of examples of SNA-rings (Smarandache non-associative rings). SNA-rings are non-associative structure on which are defined two binary operations one associative and other being non-associative and addition distributes over multiplication both from right and left. We are introducing a new concept of quasigroup rings. These are non associative structures. In our view groupoid rings and quasigroup rings are the rich source of non-associative SNA-rings without unit since all other rings happen to be either associative or non-associative rings with unit. To make this paper self contained we recollect some definitions and results which we will use subsequently. §2. Preliminaries Definition 2.1. A groupoid S such that for all a, b ∈ S there exist unique x, y ∈ S such that ax = b and ya = b is called a quasigroup. Thus a quasigroup does not have an identity element and it is also non-associative. Here is a quasigroup that is not a loop. ∗ 1 2 3 4 5 1 3 1 4 2 5 2 5 2 3 1 4 3 1 4 2 5 3 4 4 5 1 3 2 5 2 3 5 4 1 140 Arun S. Muktibodh No. 2 We note that the definition of quasigroup Q forces it to have a property that every element of Q appears exactly once in every row and column of its operation table. Such a table is called a LATIN SQUARE. Thus, quasigroup is precisely a groupoid whose multiplication table is a LATIN SQUARE. Definition 2.2. If a quasigroup (Q, ∗) contains a group (G, ∗) properly then the quasigroup is said to be Smarandache quasigroup. Example 2.1. Let Q be a quasigroup defined by the following table: ∗ a0 a1 a2 a3 a4 a0 a0 a1 a3 a4 a2 a1 a1 a0 a2 a3 a4 a2 a3 a4 a1 a2 a0 a3 a4 a2 a0 a1 a3 a4 a2 a3 a4 a0 a1 Clearly, A = {a0 , a1 } is a group w.r.t. ∗ which is a proper subset of Q. Therefore Q is a Smarandache quasigroup. Definition 2.3. A quasigroup Q is idempotent if every element x in Q satisfies x ∗ x = x. Definition 2.4. A ring (R, +, ∗) is said to be a non-associative ring if (R, +) is an additive abelian group, (R, ∗) is a non-associative semigroup (i.e. binary operation ∗ is non-associative) such that the distributive laws a ∗ (b + c) = a ∗ b + a ∗ c and (a + b) ∗ c = a ∗ c + b ∗ c for all a, b, c in R. Definition 2.5. Let R be a commutative ring with one. G be any group (S any semigroup with unit) RG (RS the semigroup ring of the semigroup S over the ring R) the group ring of the n X group G over the ring R consists of finite formal sums of the form αi gi , (n < ∞) i.e. i runs i=1 over a finite number where αi ∈ R and gi ∈ G (gi ∈ S) satisfying the following conditions: n n X X 1. αi mi = βi mi ⇔ αi = βi , for i = 1, 2, · · · , n i=1 2. n X i=1 αi mi + i=1 n X 3. ( 4. 5. n X i=1 n X αi mi )( i=1 βi mi ⇔ βi mi ) = i=1 n X i=1 n X (αi + βi)mi γk mk , mk = mi mj , where γk = P αi βi i=1 ri mi = mi ri for all ri ∈ R and mi ∈ G(mi ∈ S). n n n X X X r ri mi = rri mi for all r ∈ R and ri mi ∈ RG. RG is an associative ring with i=1 i=1 i=1 0 ∈ R acts as its additive identity. Since I ∈ R we have G = IG ⊆ RG and R.e = R ⊆ RG where e is the identity element of G. If we replace the group G in the above definition by a quasigroup Q we get RQ the quasigroup ring which will satisfy all the five conditions 1 to 5 given in the definition. But RQ Vol. 1 Smarandache quasigroup rings 141 will only be a non-associative ring without identy. As I ∈ R we have Q ⊆ RQ. Thus we define quasigroup rings as follows: Definition 2.6. For any quasigroup Q the quasigroup ring RQ is the quasigroup Q over n X the ring R consisting of all finite formal sums of the form ri qi , (n < ∞) i.e. i runs over i=1 a finite number where ri ∈ R and qi ∈ Q satisfying conditions 1 to 5 given in the definition of group rings above. Note that only when Q is a quasigroup with identity (i.e. then Q is a Loop) that the quasigroup ring RQ will be a non-associative ring with unit. Here we give examples of nonassociative quasigroup rings. Example 2.2. Let Z be the ring of integers and (Q, ∗) be the quasigroup given by the following table: ∗ 1 2 3 4 5 1 3 1 4 2 5 2 5 2 3 1 4 3 1 4 2 5 3 4 4 5 1 3 2 5 2 3 5 4 1 Clearly (Q, ∗) is a quasigroup and does not posses an identity element. The quasigroup ring ZQ is a non-associative ring without unit element. Example 2.3. Let R be the ring of reals and (Q, ∗) be the quasigroup defined by the following table: ∗ 1 2 3 4 1 1 3 4 2 2 4 2 1 3 3 2 4 3 1 4 3 1 2 4 (Q, ∗) is an idempotent quasigroup. Again RQ is a non-associative quasigroup ring without unit. Note that Rh1i, Rh2i, Rh3i, Rh4i are the subrings of RQ which are associative. Result: All quasigroup rings RQ of a quasigroup Q over the ring R are non-associative rings without unit. The smallest non-associative ring without unit is quasigroup ring given by the following example. This example was quoted by W.B.Kandasamy [2] as a groupoid ring. Example 2.4. Let Z2 = {0, 1} be the prime field of characteristic 2. (Q, ∗) be a quasigroup of order 3 given by the following table: 142 Arun S. Muktibodh ∗ q1 q2 q3 q1 q1 q2 q3 q2 q3 q1 q2 q3 q2 q3 q1 No. 2 Z2 Q is a quasigroup ring having only eight elements given by {0, q1 , q2 , q3 , q1 + q2 , q2 + q3 , q1 + q3 , q1 + q2 + q3 }. Clearly, Z2 Q is a non-associative ring without unit. This happens to be the smallest non-associative ring without unit known to us. §3. SNA-Quasigroup rings We introduce Smarandache non-associative quasigroup rings. It is true that quasigroup rings are always non-associative. We write “Smarandache non-associative quasigroup ring” only to emphasize the fact that they are non-associative. Definition 3.1. Let S be a quasigroup ring. S is said to be SNA-quasigroup ring (Smarandache non-associative quasigroup ring ) if S contains a proper subset P such that P is an associative ring under the operations of S. Example 3.1. Let Z be the ring of integers and Q be a quasigroup defined by the following table; ∗ a0 a1 a2 a3 a4 a0 a0 a1 a3 a4 a2 a1 a1 a0 a2 a3 a4 a2 a3 a4 a1 a2 a0 a3 a4 a2 a0 a1 a3 a4 a2 a3 a4 a0 a1 Clearly, A = {a0 , a1 } is group and ZQ ⊃ ZA. Thus the quasigroup ring ZQ contains an associative ring properly. Hence ZQ is an SNA-quasigroup ring. Note that Q is a Smarandache quasigroup. Example 3.2. Let R be the reals, (Q, ∗) be the quasigroup defined by the following table; ∗ 0 1 2 3 0 0 1 3 2 1 1 0 2 3 2 3 2 1 0 3 2 3 0 1 Vol. 1 Smarandache quasigroup rings 143 Then clearly RQ is an SNA-quasigroup ring as RQ ⊃ Rh0, 1i and Rh0, 1i is an associative ring. Theorem 3.1. Let Q be a quasigroup and R be any ring. Then the quasigroup ring RQ is not always an SNA-quasigroup ring. Proof. Since Q does not have an identity element, there is no guarantee that R is contained in RQ . Example 3.3. Let R be an arbitrary ring and Q be a quasigroup defined by the table; ∗ 1 2 3 4 5 1 3 1 4 2 5 2 5 2 3 1 4 3 1 4 2 5 3 4 4 5 1 3 2 5 2 3 5 4 1 Then clearly, RQ is not an SNA-quasigroup ring as the quasigroup ring RQ does not contain an associative ring. Theorem 3.2. quasigroup ring. If Q is a quasigroup with identity, then quasigroup ring RQ is SNA- Proof. Quasigroup with identity is a Loop. So, RI ⊆ RQ and R serves as the associative ring in RQ. Thus RQ is an SNA-quasigroup ring. Theorem 3.3. Let R be a ring. If Q is a Smarandache quasigroup, then quasigroup ring RQ is an SNA-quasigroup ring. . Proof. Obviously RQ is a non-associative ring. As Q is a Smarandache quasigroup Q contains a group G properly. So RQ ⊃ RG and RG is an associative ring contained in RQ. Therefore RQ is an SNA-quasigroup ring. §4. Substructure of SNA-quasigroup rings Definition 4.1. Let R be a SNA-quasigroup ring. Let S be a non-empty subset of R. Then S is said to be S-quasigroup subring of R if S itself is a ring and contains a proper subset P such that P is an associative ring under the operation of R. Example 4.1. following table: Let Z be the ring of integers. Let Q be the quasigroup defined by the 144 Arun S. Muktibodh No. 2 ∗ 1 2 3 4 5 6 7 8 1 1 2 3 4 6 5 8 7 2 2 1 4 3 5 6 7 8 3 3 4 1 2 7 8 6 5 4 4 3 2 1 8 7 5 6 5 6 5 7 8 1 2 3 4 6 5 6 8 7 2 3 4 1 7 8 7 6 5 3 4 1 2 8 7 8 5 6 4 1 2 3 Clearly the quasigroup ring ZQ is a non-associative ring. Consider the subset S = {1, 2, 3, 4} then S is a group and hence ZS is a group ring and hence also a quasigroup ring. Let P = {1, 2}. Note that ZS also contains ZP where P = {1, 2}. So, ZS is an S-quasigroup subring of SNAquasigroup ring ZQ. We have not yet been able to find a Smarandache non associative quasigroup subring for a given quasigroup ring. We think that it is not possible to obtain a subquasigroup for any quasigroup because for a quasigroup its composition table is a LATIN SQUARE. Theorem 4.1. Let R be a quasigroup ring, if R has a SNA-quasigroup subring S, then R itself is SNA-quasigroup ring. Proof. As S is an SNA-quasigroup surbring S contains an associative ring. As a result R contains an associtive ring. Thus R is an SNA-quasigroup ring. References [1] R.H.Bruck, A survey of binary systems, Springer Verlag, 1958. [2] W.B.Kandasamy, Smarandache non-associative (SNA) rings, Smarandache Notions (book series), American Research Press, 14(2004), 281-293. [3] D.S. Passman, The algebraic structure of group rings, Wiley- interscience, 1977. [4] J.S.Robinson Derek, A course in the theory of Groups, Springer Verlag, 1996. Scientia Magna Vol. 1 (2005), No. 2, 145-148 Two asymptotic formulae on the k + 1-power free numbers Shen Hong Xian Yang vocational and technical college Xian Yang, Shaanxi, P.R.China Abstract The main purpose of this paper is to study the distributive properties of k + 1-power free numbers, and give two interesting asymptotic formulae. Keywords k + 1-power free numbers; Asymptotic formula. §1. Introduction A natural number n is called a k + 1-power free number if it can not be divided by any p , where p is a prime number. One can obtain all k + 1-power free numbers by the following method: From the set of natural numbers (except 0 and 1) -take off all multiples of 2k+1 (i.e. 2k+1 , 2k+2 , · · · ). -take off all multiples of 3k+1 . -take off all multiples of 5k+1 . · · · and so on (take off all multiples of all k + 1-power primes). In reference [1], Professor F. Smarandache asked us to study the properties of the k + 1power free numbers sequence. Yet we still know very little about it. Now we define two new number-theoretic functions U (n) and V (n) as following, k+1 U (1) = 1, U (n) = Y p, p|n V (1) = 1, αr α1 1 V (n) = V (pα − 1) · · · (pαr − 1), 1 ) · · · U (pr ) = (p αr 1 where n is any natural number with the form n = pα 1 · · · pr . Obviously they are both multiplicative functions. In this paper, we shall use the analytic method to study the distribution properties of this sequence, and obtain two interesting asymptotic formulae. That is, we have the following two theorems: Theorem 1. Let A denote the set of all k + 1-power free numbers, then for any real number x ≥ 1, we have the asymptotic formula ¶ µ ³ 3 ´ X p2k−2 − 1 3x2 Y + O x 2 +ε , 1 + 2k+1 U (n) = 2 π p p + p2k − p2k−1 − p2k−2 n∈A n≤x 146 Shen Hong where ε denotes any fixed positive number and Y No. 2 denotes the product of all the prime numbers. p Theorem 2. For any real number x ≥ 1, we have the asymptotic formula µ ¶ ³ 3 ´ X x2 Y 1 p2k+1 + p2k − p − 1 + O x 2 +ε . V (n) = 1 − k+1 − 2 p p p2k+3 + p2k+1 n∈A n≤x §2. Proof of Theorems In this section, we shall complete the proof of Theorems. First we prove Theorem 1, let f (s) = 1 + X U (n) . ns n∈A n≤x From the Euler product formula [2] and the definition of U (n), we may have ¶ Yµ U (p) U (p2 ) U (pk ) f (s) = 1 + s + 2s + · · · + ks p p p p ¶ µ Y 1 1 1 = 1 + s−1 + 2s−1 + · · · + ks−1 p p p p µ ¶ Y 1 p(k−1)s − 1 = 1 + s−1 + 2s−1 (k−1)s p p (p − p(k−2)s ) p ¶ µ ζ(s − 1) Y p(k−1)s − 1 , = 1 + 2s−1 ζ(2(s − 1)) p (p + ps )(p(k−1)s − p(k−2)s ) where ζ(s) is the Riemann-zeta function. Obviously, we have the following two inequalities ¯ ¯ ∞ ¯X U (n) ¯¯ 1 ¯ |U (n)| ≤ n, , ¯ ¯< σ ¯ n ¯ σ−2 n=1 where σ > 2 is the real part of s. So by Perron formula [3] µ b ¶ Z b+iT X U (n) 1 xs x B(b + σ0 ) = f (s + s0 ) ds + O n s0 2iπ b−iT s T n≤x ¶ µ ¶ µ x log x ) + O x−σ0 H(N ) min(1, ) , +O x1−σ0 H(2x) min(1, T ||x|| 3 where N is the nearest integer to x, kxk = |x − N |. Taking s0 = 0, b = 3, T = x 2 , H(x) = x, 1 B(σ) = σ−2 , we have X n≤x 1 U (n) = 2iπ where R(s) = Z Yµ p 3+iT 3−iT 1+ 3 ζ(s − 1) xs R(s) ds + O(x 2 +ε ), ζ(2(s − 1)) s p2k−2 − 1 2k+1 p + p2k − p2k−1 − p2k−2 ¶ . Vol. 1 Two asymptotic formulae on the k + 1-power free numbers To estimate the main term 1 2iπ Z 3+iT 3−iT ζ(s − 1) xs R(s) ds, ζ(2(s − 1)) s we move the integral line from s = 3 ± iT to s = f (s) = 3 2 ± iT . This time, the function ζ(s − 1)xs R(s) ζ(2(s − 1))s 2 x has a simple pole point at s = 2 with residue 2ζ(2) R(2). So we have ÃZ Z 32 +iT Z 32 −iT Z 3−iT ! 3+iT 1 ζ(s − 1)xs + R(s)ds + + 3 3 2iπ ζ(2(s − 1))s 3−iT 3+iT 2 +iT 2 −iT µ ¶ x2 Y p2k−2 − 1 = 1 + 2k+1 . 2ζ(2) p p + p2k − p2k−1 − p2k−2 We can easily get the estimates ¯ ¯ ÃZ 3 Z 3−iT ! ¯ 1 ¯ 2 +iT ζ(s − 1)xs ¯ ¯ + R(s)ds¯ ¯ 3 ¯ ¯ 2πi ζ(2(s − 1))s 3+iT 2 −iT ¯ Z 3¯ ¯ ζ(σ − 1 + iT ) 3 x3 ¯ x3 ¯ ¿ R(s) ¯¯ dσ ¿ = x2 ¯ 3 ζ(2(σ − 1 + iT )) T T 2 and ¯ ¯ Z T ¯ ¯ 1 Z 32 −iT ζ(s − 1)xs ¯ ¯ R(s)ds¯ ¿ ¯ ¯ ¯ 2πi 3 +iT ζ(2(s − 2))s 0 2 ¯ ¯ ¯ ζ(1/2 + it) x 32 ¯ 3 ¯ ¯ ¯ ¯ dt ¿ x 2 +ε . ¯ ζ(1 + 2it) t ¯ π2 6 , then from the above we can obtain ¶ µ ³ 3 ´ X 3x2 Y p2k−2 − 1 U (n) = 2 + O x 2 +ε . 1 + 2k+1 2k 2k−1 2k−2 π p p +p −p −p Note the fact that ζ(2) = n∈A n≤x This completes the proof of Theorem 1. Now we come to prove Theorem 2. Let g(s) = 1 + X V (n) . ns n∈A n≤x From the Euler product formula [2] and the definition of V (n), we also have ¶ Yµ V (pk ) V (p) V (p2 ) g(s) = 1 + s + 2s + · · · + ks p p p p µ ¶ Y p − 1 p2 − 1 pk − 1 = 1+ s + + · · · + p p2s pks p à ! 1 Y 1 − p(k+1)(s−1) 1 − p1ks − s = 1 p −1 1 − ps−1 p ¶ µ Y (pks − 1)(ps−1 + 1) 1 . = ζ(s − 1) 1 − (k+1)(s−1) − ks p (p − p(k−1)s )p2s−1 p 147 148 Shen Hong No. 2 Now applying Perron formula [3], and the method of proving Theorem 1, we can also obtain the result of Theorem 2. This completes the proof of Theorems. References [1]F.Smarandache, Only problems, Not solutions, Chicago, Xiquan Publ. House, 1993. [2] Tom M.Apostol, Introduction to analytic number theory, New York, Springer-Verlag, 1976. [3] Pan Chengdong and Pan Chengbiao, Foundation of analytic number theory, Beijing, Science Press, 1997. Scientia Magna Vol. 1 (2005), No. 2, 149-151 An equation involving the Smarandache ceil function Ji Yongqiang Danfeng Teacher’s School, Shangluo, Shaanxi, P.R.China Abstract In this paper, we use the elementary methods to study the properties of the infinity series ∞ X 1 , where A denotes the set of all positive integers n such that the equation Sk (n) = ak (n), and s n n=1 n∈A give an interesting identity for it. Keywords Smarandache ceil function; Infinity series; Identity. §1. Introduction For any fixed positive integer k and any positive integer n, the Smarandache ceil function Sk (n) is defined as follows: Sk (n) = min{m ∈ N : n|mk }. This was introduced by Professor F.Smarandache. About this function, many scholar had studied its properties, see [1] and [2]. In [1], Ibstedt presented the following properties: (∀a, b ∈ N ) (a, b) = 1 ⇒ Sk (a · b) = Sk (a) · Sk (b), α and Sk (pα ) = pd k e , where p is a prime and dxe denotes the smallest integer greater than x. αr 1 α2 That is, Sk (n) is a multiplicative function. Therefore, if n = pα 1 p2 · · · pr is the prime power decomposition of n, then the following identity is obviously: d αr 1 α2 Sk (n) = Sk (pα 1 p2 · · · pr ) = p 1 α1 k e d p2 α2 k e d αr k e · · · pr . For any positive integer n, the Smarandache k−th power complements ak (n) is the smallest positive integer such that nak (n) is a complete k−th power. That is, ak (n) = min{l | n · l = mk , l ≥ 0, m ∈ N + }. αr 1 α2 If n = pα 1 p2 · · · pr denotes the factorization n into prime powers, then from the definition of ak (n), we know that it is also a multiplicative function, therefore α2 αr 1 ak (n) = ak (pα 1 )ak (p2 ) · · · ak (pr ). Let A denotes the set of the positive integers n such that the equation Sk (n) = ak (n). That is, A = {n ∈ N, Sk (n) = ak (n)}. In this paper, we use the elementary methods to study 150 Ji Yongqiang the properties of the infinity series No. 2 ∞ X 1 , and give an interesting identity for it. That is, we ns n=1 n∈A shall prove the following conclusion: Theorem. Let k ≥ 2 be a given positive integer. Then for any real number s > 1, we have the identity: ∞ X ζ ((k − 1)s) 1 = , s n ζ ((k 2 − 1)s) n=1 n∈A where ζ(s) is the Riemann zeta-function. Corollary 1. Taking k = 2 and s = 2 in the above theorem, then we have the identities: ∞ X 1 315 = . 2 4 n 2π n=1 n∈A Corollary 2. Taking k = 3 and s = 1 in the above theorem, then we have the identities: ∞ X 1575 1 = 6 . n π n=1 n∈A §2. Proof of the theorem In this section, we will complete the proof of the theorem. First, we define the arithmetical function b(n) as follows:   1, if n ∈ A, b(n) =  0, otherwise. αs 1 α2 Now let n = pα 1 p2 · · · ps denotes the factorization n into prime powers. Then from the introduction of this paper we know that Sk (n) and ak (n) both are multiplicative functions, so we only discuss the case of n = pi . If i = lk + n, (l ≥ 0, 0 ≤ n < k), then from the definition of Sk (n) and ak (n), we can easily get:   pl , if n = 0, i Sk (pi ) = pd k e =  pl+1 , if 0 < n < k.   1, if n = 0, ak (pi ) =  pk−n , if 0 < n < k. So Sk (pi ) = ak (pi ), if and only if l + 1 = k − n, i.e. n = k − l − 1, or i = lk + n = lk + k − l − 1 = (k − 1)(l + 1). Hence, pi = p(k−1)(l+1) , where 0 ≤ l ≤ k − 1. For any real number s > 1, it is clear that ∞ ∞ ∞ X X b(n) X 1 1 = < , ns ns ns n=1 n=1 n=1 n∈A Vol. 1 and 151 An equation involving the Smarandache ceil function ∞ ∞ X X 1 1 is convergent if s > 1. Thus is also convergent if s > 1, so from the Euler s n ns n=1 n=1 n∈A product formula (see [3]), we have ∞ ∞ X X 1 b(n) = s n ns n=1 n=1 = Yµ p n∈A = = = = = b(p) b(p2 ) 1 + s + 2s + · · · p p ! b(p(k−1)(l+1) ) 1+ p(k−1)(l+1)s p l=0 à ! k−1 Y X 1 1+ p(k−1)(l+1)s p l=0 ! à k−1 X Y 1 1 1 + (k−1)s p p(k−1)ls p l=0 à ! 1 Y 1 − pk(k−1)s 1 + (k−1)s p −1 p Y à ¶ k−1 X ζ ((k − 1)s) , ζ ((k 2 − 1)s) where ζ(s) is the Riemann zeta-function. This completes the proof of Theorem. 2 π6 Note that ζ(2) = π6 , ζ(6) = 945 and ζ(8) = corollaries. π8 9450 , we may immediately deduce the References [1] Ibstedt Surfinig, On the ocean of number-A few Smarandache notions and similar topics, Erthus University Press, New Mexico, USA. [2] Sabin Tabirca and Tatiana Tabirca, Some new results concerning the Smarandache ceil function, Smarandache notions journal, 13(2002), 30-36. [3] Tom M.Apstol, Introduction to analytic number theory, New York, Springer-Verlag, 1976. Scientia Magna Vol. 1 (2005), No. 2, 152-158 Smarandache Bcc-algebras Young Bar Jun Department of Mathematics Education, Gyeongsang National University, Chinju 660 − 701, Korea E-mail address: ybjun@gsnu.ac.kr jamjana@korea.com Abstract The notion of Smarandache BCC-algebras and Smarandache BCC-ideas is introduced. Conditions for a (special) subset to be a Smarandache BCC-ideal are given. Keywords Smarandache BCC-algebra; Smarandache BCC-ideal. §1. Introduction Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B of A which is embedded with a strong structure S. In [10], W.B.Vasantha Kandasamy studied the concept of Smarandache groupoids, subgroupoids, ideal of groupoide, semi-normal subgroupoides, Smarandache Bol groupoids and strong Bol groupoids and obtained many interesting results of congruences, and it was studied by R.Padilla [9]. In this paper, we discuss the Smarandache structure on BCC-algebras, and introduce the notion of Smarandache ideas, and investigate its properties. We give conditions for a (special) subset to be a Smarandache BCC-ideal. §2. Preliminaries BCC-algebras were introduced by Komori [7] in a connection with some problems on BCKalgebras in[11], and Dudek[1, 2] redefined the notion of BCC-algebras by using a dual form of the ordinary definition in the sense of Komori. An algebra (X; ∗, 0) of type (2, 0) is called a BCC-algebra if it satisfies the following conditions. (a1)(∀x, y, z ∈ X) (((x ∗ y) ∗ (z ∗ y)) ∗ (x ∗ z)), (a1)(∀x ∈ X) (0 ∗ x = 0), (a1)(∀x ∈ X) (x ∗ 0 = 0), (a1)(∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 =⇒ x = y). Note that every BCK-algebra is BCC-algebra, but the converse is not true. A BCC-algebra which is not a BCK-algebra is called a proper BCC-algebra. The smallest proper BCC-algebra has four elements and for every n ≥ 4 there exists at least one proper BCC-algebra [2]. A nonempty subset I of a BCC-algebra X is called a BCC-ideal of X if it satisfies the following assertions: Vol. 1 Smarandache Bcc-algebras 153 (a5) 0 ∈ I, (a6) (∀x, z ∈ X) (∀y ∈ I) ((x ∗ y) ∗ z ∈ I =⇒ x ∗ z ∈ I). Note that every BCC-algebra X satisfies the following assertions. (b1) (∀x ∈ X) (x ∗ x = 0), (b2) (∀x, y ∈ X) (x ∗ y ≤ x), (b3) (∀x, y, z ∈ X) (x ∗ y =⇒ x ≤ z ≤ y ∗ z, z ∗ y ≤ z ∗ x), where x ≤ y if and only if x ≤ y = 0. §3. Smarandache BCC-algebra We know that every proper BCC-algebra has at least four elements (see [2]), and that if X is a BCC-algebra then {0, a}, a ∈ X, is a BCC-algebra with respect to the same operation on X. Now let us consider a proper BCC-algebra X = {0, 1, 2, 3, 4} with the following Cayley table : ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 1 2 2 2 0 1 2 3 3 3 1 0 3 4 4 0 0 0 0 Table 3.1 Then {0, 1}, {0, 2}, {0, 3}, {0, 4}, {0,1,2 } and {0, 1, 3} are BCC-algebras with respect to the operation ∗ on X, and note that X does not contain BCC-algebras of order 4. Based on this result, we give the following definition. Definition 3.1. A Smarandache BCC-algebra is defined to be a BCC-algebra algebra X in which there exists a proper subset Q of X such that (¡) 0 ∈ Q, and |Q| ≥ 4, (¡¡) Q is a BCC-algebra with respect to the operation on X. Note that any proper BCC-algebra X with four elements can be Smarandache. Hence if X is a Smarandache BCC-algebra, then |x| ≥ 5. Notice that the BCC-algebra X = {0, 1, 2, 3, 4} with Table 3.1 is not a Smarandache BCC-algebra. 154 Young Bar Jun No. 2 Example 3.2. (1)Let X = {0, a, b, c, d, e} be a set with the following Cayley table: ∗ 0 a b c d e 0 0 0 0 0 0 0 a a 0 0 0 0 a b b b 0 0 a a c c b a 0 a a d d d d d 0 a e e e e e e 0 Table 3.2 Then(X; ∗, 0) is a Smarandache BCC-algebra. Note that Q = {0, a, b, c} is a BCK-algebra which is contained in X. (2)Let {X; ∗, 0} be a finite BCK-chain containing at least four elements and let c be its maximal element. Let Y = X ∪ {d} , where d 6∈ X, and define a binary operation ¯ on Y as follows:    x ∗ y, if x, y ∈ X;     0, if x ∈ X, y = d; x¯y =   d, if x = d, y = 0;     c, if x = d, y ∈ X. Then (Y ; ¯, 0) is a Smarandache BCC-algebra. (3)Let {X; ∗, 0} be a BCC-algebra containing at least four elements in which a is the small atom. Let Y = X ∪ {ω} , where ω 6∈ X,and define a binary operation ¯ on Y as follows:   x ∗ y, if x, y ∈ X;      0, if y ∈ X, x = ω; x¯y =   d, if x = ω = y;     c, if x ∈ X/{0}, y = ω. Then (Y ; ¯, 0) is a Smarandache BCC-algebra. In what follows , let X and Q denote a Smarandache BCC-algebra and non-trivial BCCalgebra which is properly contained in X , respectively, unless otherwise specified. Definition 3.3. A nonempty subset I of X is called a Smarandache BCC-ideal of X related to Q if it satisfies: (c1) 0 ∈ I, (c2) (∀x, z ∈ Q) (∀y ∈ I) ((x ∗ y) ∗ z ∈ I =⇒ x ∗ z ∈ I). If I is a Smarandache BCC-ideal of X related to every non-trivial BCC-algebra contained in X, we simply say that I is a Smarandache BCC-ideal of X. Example 3.4. (1)Let X = {0, a, b, c, d, e} be the Smarandache BCC-algebra described in Example 3.2(1). Then I = {0, a} and J = {0, a, b, c, d} are Smarandache BCC-ideals of X related to Q = {0, a, b, c}. Vol. 1 Smarandache Bcc-algebras 155 Proposition 3.5. Every Smarandache BCC-ideals of X related to Q satisfies: (c3) (∀x ∈ Q) (∀a ∈ I) (x ∗ a ∈ I =⇒ x ∈ I), (c4) (∀x ∈ Q) (∀a ∈ I) (a ∗ x ∈ I), (c5) (∀x ∈ Q) (∀a, b ∈ I) (x ∗ ((x ∗ a) ∗ b ∈ I)). Proof. (c3) Taking z = 0 and y = a in (c2) and using (a3) induces the desired implication. (c4) For every x ∈ Q and a ∈ I, we have (a ∗ a) ∗ x = 0 ∗ x = 0 ∈ I and so a ∗ x ∈ I by (c2). (c5) Let x ∈ Q and a, b ∈ I. Then (x ∗ a) ∗ (a ∗ x) = 0 ∈ I, and so x ∗ (x ∗ a) ∈ I by (c2). Since ((x ∗ b) ∗ ((x ∗ a) ∗ b)) ∗ (x ∗ (x ∗ a)) = 0 ∈ I, it follows from (c3) that (x ∗ b) ∗ ((x ∗ a) ∗ b) ∈ I so from (c2) that x ∗ ((x ∗ a) ∗ b) ∈ I. Corollary 3.6. For every Smarandache BCC-ideal I of X related to Q, the following implication is valid: (∀x ∈ Q)(∀a ∈ I)(x ≤ a =⇒ x ∈ I). Corollary 3.7. Let I be a Smarandache BCC-ideal I of X related to Q . Then (∀x ∈ Q)(∀a, b ∈ I)(x ∗ a ≤ b =⇒ x ∈ I). Theorem 3.8. Let Q1 and Q2 be non-trivial BCK-algebras which are properly contained in X such that Q1 ⊂ Q2 . Then every Smarandache BCC-ideal of X related to Q2 is a Smarandache BCC-ideal of X related to Q1 . Proof. Straightforward. Corollary 3.9. If Q is the largest BCK-algebra which is properly contained in X, then every Smarandache BCC-ideal of X related to Q is a Smarandache BCC-ideal of X. The converse of Theorem 3.8 is not true in general as seen in the following example. Example 3.10. Consider a Smarandache BCC-algebra X = {0, 1, 2, 3, 4, 5} with the following Cayley table: ∗ 0 1 2 3 4 5 0 0 0 0 0 0 0 1 1 0 0 0 0 1 2 2 1 0 0 0 1 3 3 1 1 0 1 1 4 4 1 1 1 0 1 5 5 5 5 5 5 0 Table 3.3 Note that Q1 := {0, 1, 2, 3} and Q2 := {0, 1, 2, 3, 4} are BCK-algebras. Then the set Q1 is a Smarandache BCC-ideal of X related to Q1 , but not Q2 . In fact, we know that (4∗2)∗0 = 1 ∈ Q1 and 4 ∗ 0 = 4Q1 . Remark 3.11. Note that every BCC-ideal of X is a Smarandache BCC-ideal of X, but the converse is not valid. Example 3.10 shows that there exists a BCC-algebra Q of order 156 Young Bar Jun No. 2 n ≥ 4 which is properly contained in a Smarandache BCC-algebra X such that a Smarandache BCC-ideal of X related to Q is not a BCC-ideal of X. We provide convide conditions for a subset to be a Smarandache BCC-ideal. Theorem 3.12. If I is a subset of Q that satisfies conditions (c1) and (c3), then I is a Smarandache BCC-ideal of X related to Q. Proof. Let x, y ∈ Q and a ∈ I be such that (x ∗ a) ∗ y ∈ I. Since a ∈ I ⊆ Q and Q is a BCK-algebra, it follows that (x ∗ y) ∗ a = (x ∗ a) ∗ y ∈ I. So from (c3) that x ∗ y ∈ I. Hence I is a Smarandache BCC-ideal of X related to Q. Theorem 3.13. If a nonempty subset I of X satisfies conditions (c1) and (c5), then I is a Smarandache BCC-ideal of X related to Q. Proof. Let x, y ∈ Q and a ∈ I be such that (x ∗ a) ∗ y ∈ I. Taking b = 0 in (c5) and using (a3), we have (x ∗ x) ∗ a ∈ I. It follows from (a3), (a1) and (c5) that x ∗ y = (x ∗ y) ∗ 0 = (x ∗ y) ∗ (((x ∗ y) ∗ ((x ∗ a) ∗ y)) ∗ (x ∗ (x ∗ a))) ∈ I. Thus I is a Smarandache BCC-ideal of X related to Q. Theorem 3.14. Let H be a BCC-subalgebra of X. Then H is a Smarandache BCC-ideal of X related to Q if and only if it satisfies: (∀x ∈ H)(∀y, z ∈ Q)((y ∗ x) ∗ z ∈ H =⇒ y ∗ z ∈ H). Proof. Straightforward. Given an element ω ∈ X\{0}, consider the set [0, ω] := {x ∈ Xkx ≤ ω}. which is called the initial segment of ω[5]. Obviously, 0 ∈ [0, ω] for all x ∈ ω. Since x ≤ ω is equivalent to xω = 0, the initial segment of ω is defacto the left annihilator of ω. In general, [0, ω] is not a Smarandache BCC-ideal of X, but it is a subalgebra. For example, let X be the Smarandache BCC-algebra in example 3.2(1). Then [0, e] = {0, e} is not a Smarandache BCC-ideal of X related to Q = {0, a, b, c} since (b ∗ e) ∗ d = 0 ∈ [0, e], but b ∗ d = a[0, e]. Theorem 3.15. For every c ∈ X\{0}, if the inequality (∀x ∈ Q)(x ∗ ((x ∗ c) ∗ c) ≤ c holds, then [0, c] is a Smarandache BCC-ideal of X related to Q. Proof. Let x ∈ Q. If b ∈ [0, c], then b ≤ c and hence (x ∗ c) ∗ c ≤ (x ∗ c) ∗ b by (b3). It follows from (b3) and assumption that x ∗ ((x ∗ c) ∗ b) ≤ x ∗ ((x ∗ c) ∗ c) ≤ c. Now if a ∈ [0, c], then x ∗ c ≤ x ∗ a, and so x ∗ ((x ∗ a) ∗ b) ≤ x ∗ ((x ∗ c) ∗ b) ≤ c. This shows that x ∗ ((x ∗ a) ∗ b) ∈ [0, c]. Applying Theorem 3.13, we conclude that [0, c] is a Smarandache BCC-ideal of X related to Q. Vol. 1 Smarandache Bcc-algebras 157 Theorem 3.16. The initial segment [0, c], where c ∈ X/{0}, is a Smarandache BCC-ideal of X related to Q if and only if the implication (∀x, y ∈ Q)((x ∗ c) ∗ y ≤ c =⇒ x ∗ z ∈ I) is valid. Proof. Let ∀x, y ∈ Q and a ∈ [0, c] be such that (x ∗ a) ∗ y ∈ [0, c]. Then a ≤ c and (x ∗ a) ∗ y ≤ c. The inequality z ≤ c implies that (x ∗ c) ∗ y ≤ (x ∗ a) ∗ y ≤ c so from hypothesis that x ∗ y ≤ c, that is, x ∗ y ∈ [0, c] is a Smarandache BCC-ideal of X related to Q. Conversely assume that [0, c], c ∈ X\{0}, is a Smarandache BCC-ideal of X related to Q and let ∀x, y ∈ Q be such that (x ∗ c) ∗ y ≤ c. Then (x ∗ c) ∗ y ∈ [0, c]. Since [0, c] is a Smarandache BCC-ideal of X related to Q and c ∈ [0, c], it follows from (c2) that x ∗ y ∈ [0, c] so that x ∗ y ≤ c. This completes the proof. Corollary 3.17. If [0, c], c ∈ X/{0}, is a Smarandache BCC-ideal of X related to Q, then (∀x, y ∈ Q)((x ∗ c) ≤ c =⇒ x ≤ c). Theorem 3.18. For every c ∈ X/{0}, if the equality (∀x, y ∈ Q)(((x ∗ c) ∗ y) ∗ c = (x ∗ y) ∗ c) is valid, then [0, c] is a Smarandache BCC-ideal of X related to Q. Proof. Let ∀x, y ∈ Q and a ∈ [0, c] be such that (x ∗ a) ∗ y ∈ [0, c]. Then a ≤ c and (x ∗ a) ∗ y ≤ c. It follows that (x ∗ y) ∗ c = ((x ∗ c) ∗ y) ∗ c ≤ ((x ∗ a) ∗ y) ∗ c ≤ c ∗ c = 0, so that (x ∗ y) ∗ c = 0, i.e., x ∗ y ≤ c. Hence x ∗ y ∈ [0, c] and therefore [0, c] is a Smarandache BCC-ideal of X related to Q. Acknowledgements This paper was supported by Korea Research Foundation Grant (KRF-2003-005-C00013). References [1] W.A.Dudek, The number of subalgebras of finite BCC-algebras, Bull.Inst.Math, Academia Sinica, 20(1992), 129-136. [2] W.A.Dudek, On proper BCC-algebras, Bull.Inst.Math, Academia Sinica 20(1992), 137150. [3] W.A.Dudek, On constructions of BCC-algebras,selected Papers on BCK and BCIalgebras, Xi‘an, China, Shaanxi Scientific and Technological Press, 1(1992), 93-96. [4] W.A.Dudek and X.H.Zhang, On ideals and congruences in BCC-algebras, Coravica, 4(1998), 21-29. 158 Young Bar Jun No. 2 [5] W.A.Dudek and X.H.Zhang, Initial segments in BCC-algebras, Math.Moreavica, 4(2000), 27-34. [6] J.Hao, Ideal theory of BCC-algebras, Sci.Math, 3(1998), 378-381. [7] Y.Komori, The class of CC-algebras is not a variety, Math. Japon, 29(1984), 391-394. [8] J.Meng and Y.B.Jun, BCC-algebras, Kyungmoonsa Ca.Seoul, Korea, (1994). [9] R.Padilla, Smarandache algebraic structures, Bull Pure Appl.Sci.Delhi, 17E(1998), 119121; http://www.gallup.unm.edu/smarandache/alg-s-tx.txt. [10] W.B.Vasantha Kandasamy, Smarandache groupoids, http://www.gallup.unm.edu/smarandache/Groupoids.pdf. [11] A.Wronski, BCK-algebras do not form a variety, Math. Japon, 28(1983), 211-213. Scientia Magna Vol. 1 (2005), No. 2, 159-161 On the asymptotic property for Smarandache additive factorial complements Yang Mingshun and Yang Qianli Department of Mathematics, Weinan Teacher’s College Weinan, Shaanxi, P.R.China Abstract The main purpose of this paper is to study the mean value properties of the Smarandache additive factorial complements, and give an interesting asymptotic formula for it. Keywords Smarandache additive factorial complements; Asymptotic formula. §1. Introduction For any positive integer n, the Smarandache factorial complements quotients of n, denoted by a(n), is defined as follows c(n) = min{k|nk = m!, k ≥ 0, m ∈ N + }. In problem 45 of reference [1], professor F.Smarandache asked us to study the properties of the factorial complements. Analogously, we can define the Smarandache additive factorial complements a(n): a(n) = min{k|n + k = m!, k ≥ 0, m ∈ N + }. About arithmetical properties of the this sequence, it seems that none had studied it before. In this paper, we use the elementary method to study the mean value properties of the Smarandache additive factorial complements, and obtain an interesting asymptotic formula for it. That is, we shall prove the following: Theorem. For any real number x ≥ 1, we have the asymptotic formula µ 2 ¶ X 1 ln2 x ln x ln ln ln x = +O . a(n) + 1 2 ln ln x (ln ln x)2 n≤x §2. Proof of the theorem In this section, we will complete the proof of the theorem. For any real number x, let positive integer m satisfy M ! ≤ x < (M + 1)!. (1) 160 Yang Mingshun and Yang Qianli No. 2 Taking the logistic computation on both sides of the inequality, we get M X ln i ≤ ln x < i=1 M +1 X ln i. i=1 And then using the Euler’s summation formula we have M X ln i = M ln M − M + O(ln M ) (2) ln i = M ln M − M + O(ln M ). (3) i=1 and M +1 X i=1 Combining (2) and (3), we can easily deduce that ln x = M ln M − M + O(ln M ). So we get ln x + O(1). ln M − 1 Similarly, taking logistic computation on both sides we have M= ln M = ln ln x + O(ln ln M ) and ln x +O M= ln ln x µ (4) ln x ln ln ln x (ln ln x)2 ¶ According to the definition of a(n) and (1), we can write   X X X 1 1  + = a(n) + 1 a(n) + 1 n≤x 1≤m≤M −1 X = m!≤n<(m+1)! m·m! X 1≤m≤M −1 i=1 µ X = 1 + i X n≤x−M !+1 . (5) X M !≤n≤x 1 a(n) + n 1 n µ ln(m · m!) + γ − 1 + O 1≤m≤M −1 1 m · m! ¶¶ + O (ln(M · M !)) , where γ is the Euler’s constant. Now combining the Stirling Formula (see reference [2]), we can get X n≤x 1 a(n) + 1 = X 1≤m≤M −1 = X ln m! + X ln m + O(M ln M ) 1≤m≤M −1 m ln(m + 1) − 1≤m≤M −1 X m + O(M ln M ). 1≤m≤M −1 Applying Abel’s identity (see reference [3]), we have X n≤x 1 1 3 = M 2 ln M − M 2 + O(M ln M ). a(n) + 1 2 4 Vol. 1 On the asymptotic property for Smarandache additive factorial complements 161 So from (4) and (5), we obtain X n≤x 1 ln2 x = +O a(n) + 1 2 ln ln x µ ln2 x ln ln ln x (ln ln x)2 ¶ . This completes the proof of Theorem. References [1] F.Smarandache, Only problems, not solutions, Chicago, Xiquan Publ. House, 1993. [2] Pan Chengdong and Pan Chengbiao, Foundation of analytic number theory, Beijing, Science publishing house, 1999, pp49. [3] Tom M.Apostol, Introduction to analytic number theory, New York, Springer-Verlag, 1976, pp77. Scientia Magna Vol. 1 (2005), No. 2, 162-166 The Smarandache minimum and maximum functions József Sándor Babes-Bolyai University of Cluj, Romania Abstract This papers deals with the introduction and preliminary study of the Smarandache minimum and maximum functions. Keywords Smarandache minimum and maximum functions; arithmetical properties. 1. Let f : N ∗ −→ N be a given arithmetic function and A ⊂ N a given set. The arithmetic function FfA (n) = min{k ∈ A : n | f (k)} (1) has been introduced in [4] and [5]. For A = N, f (k) = k! one obtains the Smarandache function; For A = N ∗ , A = p = {2, 3, 5, ·} = set of all primes, one obtains a function P (n) = min{k ∈ P : n | k!} (2) For the properties of this function, see [4] and [5]. The “dual” function of (1) has been defined by GA g (n) = max{k ∈ A : g(k) | n}, (3) where g : N ∗ −→ N is a given function, and A ∈ N is a given set. Particularly, forA = N ∗ , g(k) = k!, one obtains the dual of the Smarandache function, S∗ (n) = max{k ≥ 1 : k! | n} (4) For the properties of this function, see [4] and [5]. F.Luca [3], K.Atanassov [1] and L.le [2] have proved in the affirmative a conjecture of the author. For A = N ∗ and f (k) = g(k) = ϕ(k) in (1), resp.(3) one obtains the Euler minimum, resp. maximum-function, defined by E(n) = min{k ≥ 1 : n | ϕ(k)}, (5) Vol. 1 The Smarandache minimum and maximum functions E∗ (n) = max{k ≥ 1 : ϕ(k) | n} 163 (6) For the properties of these function, see [6]. When A = N ∗ , f (k) = d(k) =number of divisors of k, one obtains the divisor minimum function (see [4], [5] and [7]) D(n) = min{k ≥ 1 : n | d(k)}. (7) It is interesting to note that the divisor maximum function (i.e., the “ dual” of D(n)) given by D∗ (n) = max{k ≥ 1 : d(k) | n} (8) is not well defined! Indeed, for any prime p one has d(pn−1 ) = n | n and pn−1 is unbounded as p −→ ∞. For a finite set A, however D∗A (n) does exist. On one hand, it has been shown in [4] and [5] that X (n) = min{k ≥ 1 : n | σ(k)} (9) (denoted there by Fσ (n)) is well defined. (Here σ(k) denotes the sum of all divisors of k). The dual of the sum-of-divisors minimum function is X ∗ (n) = max(k ≥ 1 : σ(k) | n}) (10) P Since σ(1) = 1 | n and σ(k) ≥ k, clearly ∗ (n) ≤ n, so this function is well defined (see [8]). 2. The Smarandache minimum function will be defined for A = N ∗ , f (k) = S(k) in (1). Let us denote this function by Smin : Smin (n) = min{k ≥ 1 : n | S(k)} (11) Let us assume that S(1) = 1, i. e., S(n) is defined by (1) for A = N ∗ , f (k) = k! : S(n) = min{k ≥ 1 : n | k!} (12) Otherwise (i.e.when S(1) = 0) by n | 0 for all n, by (11) for one gets the trivial function Smin (n) = 0. By this assumption, however, one obtains a very interesting (and difficult) function smin given by (11). Since n | S(n!) = n, this function is correctly defined. The Smarandache maximum function will be defined as the dual of Smin : Smax (n) = max(k ≥ 1 : S(k) | n}. (13) We prove that this is well defined. Indeed, for a fixed n, there are a finite number of divisors of n, let i | n be one of them. The equation S(k) = i (14) is well-known to have a number of d(i!) − d((i − 1)!) solutions, i. e., in a finite number. This implies that for a given n there are at most finitely many k with S(k) | k, so the maximum in (13) is attained. 164 József Sándor No. 2 Clearly Smin (1) = 1, Smin (2) = 2, Smin (3) = 3, Smin (4) = 4, Smin (5) = 5, Smin (6) = 9, Smin (7) = 7, Smin (8) = 32, Smin (9) = 27, Smin (10) = 25, Smin (11) = 11, etc, which can be determined from a table of Smarandache numbers: n 1 2 3 4 5 6 7 8 9 10 11 12 13 S(n) 1 2 3 4 5 3 7 4 6 5 11 4 13 n 14 15 16 17 18 19 20 21 22 23 24 25 S(n) 7 5 6 7 6 19 5 7 11 23 4 10 We first prove that: Theorem 1. Smin (n) ≥ n for all n ≥ 1, with equality only for n = 1, 4, p(p = prime) (15) Proof. Let n | S(k). If we would have k < n, then since S(k) ≤ k < n we should get S(k) < n, in contradiction with n | S(k). Thus k ≥ n, and taking minimum, the inequality follows. There is equality for n = 1 and n = 4. Let now n > 4. If n = p =prime, then p | S(p) = p, but for k < p, p † S(k). Indeed, by S(k) ≤ k < p this is impossible. Reciprocally, if min{k ≥ 1 : n | S(k)} = n, then n | S(n), and by S(n) ≤ n this is possible only when S(n) = n, i. e., when n = 1, 4, p(p = prime). Theorem 2. For all n ≥ 1, Smin (n) ≤ n! ≤ Smax (n) (16) Proof. Since S(n!)=n, definition (11) gives the left side of (16), while definition (13) gives the right side inequality. X X 1 1 Corollary. The series is divergent, while the series is convergent. Smin (n) Smax (n) n≥1 n≥1 X 1 X 1 ≤ = e − 1 by (16), this series is convergent. On the Proof. Since Smax (n) n! n≥1 other hand, n≥1 X n≥1 1 Smin (n) ≥ X 1 p Smin (p) = X1 p p = +∞, so the first series is divergent. Theorem 3. For all primes p one has Smax (p) = p! (17) Proof. Let S(k) | p. Then S(k) = 1 or S(k) = p. We prove that if S(k) = p, then k ≤ p!. Indeed, this follows from the definition (12), since S(k) = min{m ≥ 1 : k | m!} = p implies k | p!, so k ≤ p!. Therefore the greatest value of k is k = p!, when S(k) = p | p. This proves relation (17). Theorem 4. For all primes p, Smin (2p) ≤ p2 ≤ Smax (2p) (18) Vol. 1 The Smarandache minimum and maximum functions 165 and more generally; for all m ≤ p, Smin (mp) ≤ pm ≤ Smax (mp) (19) Proof. (19) follows by the known relation S(pm ) = mp if m ≤ p and the definition (11), (13). Particularly, for m = 2, (19) reduces to (18). For m = p, (19) gives Smin (p2 ) ≤ pp ≤ Smax (p2 ) (20) This case when m is also an arbitrary prime is given in. Theorem 5. For all odd primes p and q, p < q one has Smin (pq) ≤ q p ≤ pq ≤ Smax (pq) (21) (21) holds also when p = 2 and q ≥ 5. Proof. Since S(q p ) = pq and S(pq ) = qp for primes p and q, the extreme inequalities of (21) follow from the definition (11) and (13). For the inequality q p < pq remark that this is equivalent to f (p) > f (q), where f (x) = lnxx (x ≥ 3). x Since f 0 (x) = 1−ln = 0 ⇔ x = e immediately follows that f is strictly decreasing for x2 x ≥ e = 2.71· From the graph of this function, since ln22 = ln44 we get that ln 2 ln 3 < , 2 3 but ln q ln 2 > 2 q for q ≥ 5. Therefore (21) holds when p = 2 and q ≥ 5. Indeed, f (q) ≤ f (5) < f (4) = f (2). Remark. For all primes p, q Smin (pq) ≤ min{pq , q p } (22) Smax (pq) ≥ max{pq , q p }. (23) and For p = q this implies relation (21). Proof. Since S(q p ) = S(pq ) = pq, one has Smin (pq) ≤ pq , Smin (pq) ≤ q p , Smax (pq) ≤ pq , Smax (pq) ≤ q p . 166 József Sándor No. 2 References [1] K.Atanassov, Remark on Jozsef Sandor and Florian Luca’s theorem, C. R. Acad. Bulg. Sci., 55(10)(2002), 9-14. [2] M.Le, A conjecture concerning the Smarandache dual function, Smarandache Notions J., 14(2004), 153-155. [3] F.Luca, On a divisibility property involving factorials, C. R. Acad. Bulg. Sci., 53(6)(2000), 35-38. [4] J.Sandor, On certain generalizations of the Smarandache function, Notes Number Theorem Disci. Math., 5(2)(1999),41-51. [5] J.Sandor, On certain generalizations of the Smarandache function, Smarandache Notions Journal, 11(2000), 202-212. [6] J.Sandor, On the Euler minimum and maximum functions(to appear). Scientia Magna Vol. 1 (2005), No. 2, 167-172 Some properties of the Pseudo-Smarandache function Richard Pinch 2 ELDON Road, Cheltenham, Glos GL52 6TU, U.K. E-mail address: rgep@chalcedon.demon.co.uk Abstract Charles Ashbacher [1] has posed a number of questions relating to the pseudo-smarandache function Z(n). In this note we show that the ratio of consecutive values Z(n+1)/Z(n) and Z(n−1)/Z(n) are unbounded; that Z(2n)/Z(n) is unbounded; that n/Z(n) takes every integer value infinitely often; P and that the n 1/Z(n)α is convergent for any α > 1. §1. Introduction We defined the m-th triangular number T (m) = pseudo-Smarandache function Z(n) by m(m+1) . 2 Kashihara [2] has defined the Z(n) = min{m : n|T (m)}. Charles Ashbacher [1] has posed a number of questions relating to pseudo-Smarandache function Z(n). In this note, we show that the ratio of consecutive values Z(n)/Z(n−1) and Z(n)/Z(n+1) are unbounded; that Z(2n)/Z(n) is unbounded; and that n/Z(n) takes every integer value P infinitely often. He notes that the series n 1/Z(n)α is divergent for α = 1 and asks whether it is convergent for α = 2. He further suggests that the least value α for which the series converges “ may never be known ” . We resolve this problem by showing that the series converges for all α > 1. §2. Some Properties of t he Pseudo-Smarandache Function We record some elementary properties of the funtion Z. Lemma 1.(1) If n ≥ T (m), then Z(n) ≥ m, Z(T (m)) = m. √ (2)For all n we have n < Z(n). (3)Z(n) ≤ 2n − 1, and if n is odd, then Z(n) ≤ n − 1. (4)If p is an odd prime dividing n, then Z(n) ≥ p − 1. (5)Z(2k ) = 2k+1 − 1. (6)If p is an odd prime, then Z(pk ) = pk − 1 and Z(2pk ) = pk − 1 or pk according as pk ≡ 1 or 3 mod 4. We shall make use of Dirichlet’s Theorem on primes in arithmetic progression in the following form. 168 Richard Pinch No. 2 Lemma 2. Let a, b be coprime integers. Then the arithmetic progression a + bt is prime for infinitely many values of t. §3. Successive Values of the Pseudo-Smarandache Function Using the properties (3) and (5), Ashbacher observed that |Z(2k ) − Z(2k − 1)| > 2k and so the difference between the consecutive of Z is unbounded. He asks about the ratio of consecutive values. Theorem 1. For any given L > 0 there are infinitely many values of n such that Z(n + 1)/Z(n) > L, and there are infinitely many values of such that Z(n − 1)/Z(n) > L . Proof. Choose k ≡ 3 mod 4, so that T (k) is even and (k + 1)|(m + 1). There are satisfied if m ≡ k mod k(k + 1), that is , m = k + k(k + 1)t for some t. We have m(m + 1) = k(1 + (k + 1)t)(k + 1)(1 + kt), so that if n = k(k + 1)(k + 1)(1 + kt)/2, we have n|T (m). Now consider n + 1 = T (k) + 1 + kT (k)t. We have k|T (k), so T (k) + 1 is coprime to both k and T (k). Thus the arithmetic progression T (k) + 1 + kT (k)t has initial term coprime to its increment and by Dirichlet’s Theorem contains infinitely many primes. We find that there are infinitely many values of t for which n + 1 is prime and so Z(n) ≤ m = k + k(k + 1)t and Z(n + 1) = n = T (k)(1 + kt). Hence n T (k) + kT (k)t k Z(n + 1) ≥ = > . Z(n) m k + 2T (k)t 3 A similar argument holds if we consider the arithmetic progression T (k) − 1 + kT (k)t. We then find infinitely many values of t for which n − 1 is prime and Z(n − 1) n−2 T (k) − 2 + kT (k)t k ≥ = > . Z(n) m k + 2T (k)t 4 The Theorem follows by taking k > 4L. We note that this Theorem, combined with Lemma 1(2) , given another proof of the result that the differences of consecutive values is unbounded. §4. Divisibility of the Pseudo-Smarandache Function Theorem 2. For any integer k ≥ 2 , the equation n/Z(n) = k has infinitely many solutions n. Proof. Fix an integer k ≥ 2. Let p be a prime ≡ −1 mod2k and put p + 1 = 2kt. Put n = T (p)/t = p(p + 1)/2t = pk. Then n|T (p) so that Z(n) ≤ p . We have p|n, so Z(n) ≥ p − 1; That is, Z(n) must be either p or p − 1. Suppose, if possible, that it is the latter. In this case we have 2n|p(p + 1) and 2n|(p − 1)p, so 2n divides p(p + 1) − (p − 1) = 2p; but this is impossible since k > 1 and so n > p. We conclude that Z(n) = p and n/Z(n) = k as required. Further, for any given value of k there are infinitely many prime values of p satisfying the congruence condition and infinitely many values of n = Y (p) such that Z/Z(n) = k. Vol. 1 Some properties of the pseudo-Smarandache function 169 §5. Another Divisibility Question Theorem 3. The ratio Z(2n)/Z(n) is not bounded above. Proof. Fix an integer k, let p ≡ −1mod 2k be prime and put n = T (p). Then Z(n) = p. Consider Z(2n) = m. We have 2k p|p(p + 1) = 2n and this divides m(m + 1)/2. We have m = ² mod p and m ≡ δ mod 2k+1 where each of ², δ can be either 0 or −1. Let m = pt + ². Then m ≡ ² − t ≡ δ mod 2k . This implies that either t = 1 or t ≥ 2k − 1. Now if t = 1 then m ≤ p and T (m) ≤ T (p) = n, which is impossible since 2n ≤ T (m). Hence t ≥ 2k − 1. Since Z(2n)/Z(n) = m/p > t/2, we see that the ratio Z(2n)/Z(n) can be made as large as desired. §6. Convergence of A Series P Ashbacher observes that the series n 1/Z(n)α diverges for α = 1 and asks whether it converges for α = 2 . Lemma 3. n X log n ≤ 1/Z(n)α ≤ 1 + log n; m=1 n X log m 1 1 (log n)2 − 0.257 ≤ ≤ (log n)2 + 0.110, 2 m 2 m=1 for n ≥ 4. Proof. For the first part, we have 1 ≤ m 1 m ≤ Z m 1 t m−1 ≤ 1 m−1 for t ∈ [m − 1, m]. Integrating, 1 1 dt ≤ t m−1 Summing, Z n n n X X 1 1 1 ≤ dt ≤ m t m −1 1 2 2 That is , n X 1 ≤ 1 + log n m 1 and log n ≤ n−1 X 1 1 m The result follows. For the second part, we similarly have log m/m ≤ log t/t ≤ log(m − 1)/(m − 1), for t ∈ [m − 1, m] when m ≥ 4, since log x/x is monotonic decreasing for x ≥ e. Integrating, log m ≤ m Z m m−1 log t m−1 dt ≤ . t m 170 Richard Pinch No. 2 Summing, Z n X log m m 4 n ≤ 3 n Xm−1 log t dt ≤ . t m 4 That is, n X log m m 1 ≤ ≤ log 2 log 3 − 2 3 − 1 1 (log n)2 − (log 3)2 2 2 n X log m 1 m − log n log 2 − . n 2 We approximate the numerical values log 2 log 3 1 + − (log 3)2 < 0.110 2 3 2 and log 2 1 − (log 3)2 > −0.257 2 2 to obtain the result. Lemma 4. Let d(m) be the function which counts the divisors of m. For n ≥ 2 we have n X d(m)/m < 7(log n)2 . m=1 Proof. We verify the assertion numerically for n ≤ 6. Now assume that n ≥ 8 > e2 , we have n n X X X 1 X X 1 d(m) = = m m de m=1 m=1 de=m = d≤n de≤n X1 X 1 X1 ≤ (1 + log(n/d)) d e d d≤n e<n/d d≤n 1 ≤ (1 + log n)2 − (log n)2 + 0.257 2 1 = 1.257 + 2 log n + (log n)2 2 1 4 log n 2 log n ) + 2 log n( ) + (log n)2 < ( 3 2 2 2 < 2(log n)2 Lemma 5. Fix an integer t ≥ 5. Let et > Y > e(t−1)/2 . The number of integers n with e > n > et such that Z(n) ≤ Y is at most 196Y t2 . Proof. Consider such an n with m = Z(n) ≤ Y . Now n|m(m + 1), say k1 n1 = m and k2 n2 = m + 1, with n = n1 n2 . Thus k = k1 k2 = m(m + 1)/n and k1 n1 ≤ Y . The value t−1 Vol. 1 Some properties of the pseudo-Smarandache function 171 of k is bounded below by 2 and above by m(m + 1)/n ≤ 2Y 2 /et−1 = K, say. Given a pair (k1 , k2 ), the possible values of n1 are bounded above by Y /k1 and must satisfy the congruence condition k1 n1 + 1 ≡ 0 modulo k2 : there are therefore at most Y /k1 k2 + 1 such values. Since Y /k ≥ Y /K = et−1 /2Y > 1/2e, we have Y /k + 1 < (2e + 1)Y /k < 7Y /k. Given values for P k1 , k2 and n1 , the value of n2 is fixed as n2 = (k1 n1 + 1)/k2 . There are thus at most d(k) P possible pairs (k1 , k2 ) and hence at most 7Y d(k)/k possible quadruples (k1 , k2 , n1 , n2 ). We have K > 2, so that the previous Lemma applies and we can deduce that the number of values of n satisfying the given conditions is most 49Y (logK)2 . Now K = 2Y 2 /et−1 < 2et+1 so log K < t + 1 + log 2 < 2t. This establishes the claimed upper bound of 196Y t2 . Theorem 4. Fix 12 < β < 1 and integer t ≥ 5. The number of integers n with et−1 < n < et , such that Z(n) < nβ is at most 196t2 eβt . Proof. We apply the previous result with Y = eβt . The conditions of β ensure that the previous Lemma is applicable and the upper bound on the number of such n is 196t2 eβt as claimed. Theorem 5. The series ∞ X 1 α Z(n) n=1 is convergent for any α > √ 2. Proof. We note that if α > 2 then f rac1Z(n)α < n1α and the series is convergent . So we q √ may assume 2 < α < 2 . Fix β with α1 < β < α2 . We have 12 < β < 12 < α2 . We split the positive integers n > e4 into two classes A and B. We let class A be the union of the At where, for postive integer t ≥ 5 we put into class At those integers n such that et−1 < n < et for integer t and Z(n) ≤ nβ . All values of n with Z(n) > nβ we put into class 1 B. We consider the sum of Z(n) α over each of the two classes. Since all terms are positive, it is sufficient to prove that each series separately is convergent. Firstly we observe that for n ∈ B, we have summed over the class B is convergent. 1 Z(n)α < 1 nαβ and since αβ > 1 the series Consider the elements n of At : so for such n we have et−1 < n < et and Z(n) < nβ . By the previous result, the number of values of n satisfying these conditions is at most 196t2 eβt . √ For n ∈ At , we have Z(n) > n, so 1/Z(n)α ≤ 1/nα/2 < 1/eα(t−1)/2 . Hence the sum of the √ P 1 2 α/2 (β−α/2)t subseries n ∈ At Z(n) e . Since β < α/2 for α > 2 , the sum α is at most 196t e over all t of these terms is finite. √ P 1 We conclude that Z(n) 2. α is convergent for any α > Theorem 6. The series ∞ X 1 α Z(n) n=1 is convergent for any α > 1. proof. We fix β0 = 1 > β1 > · · · > βr = 12 with βj < αβj+1 for 0 ≤ j ≤ r − 1. We defined a partition of the integers et−1 < n < et into classes Bt and Ct (j), 1 ≤ j ≤ r − 1. Into Bt place those n with Z(n) > nβ1 . Into Ct (j) place those n with nβj+1 < Z(n) < nβj . Since βr = 21 we see that every n with et−1 < n < et is placed into one of the classes. 172 Richard Pinch No. 2 The number of elements in Ct (j) is at most 196t2 eβj t and so X n∈Ct (j) 1 < 196t2 eβj t e−βj α(t−1) = 196t2 eβj+1 α e(βj −αβj+1 )t . Z(n)α For each j we have βj < αβj+1 so each sum over t converges. The sum over the union of the Bt is bounded above by X n 1 , nαβ1 which is convergent since αβ1 > β0 = 1. P∞ 1 We conclude that n=1 Z(n) α is convergent. References [1] Charles Ashbacher, Pluckings from the tree of Smarandache sequences and functions, American Research Press, 1998. http://www.gallup.unm.edu/ smarandache/ashbacher-pluckings.pdf. [2] K.Kashihare, Comments and topics on Smarandache notions and problems, Erhus University Press,Vall, AZ, USA, 1996. Scientia Magna Vol. 1 (2005), No. 2, 173-175 An equation involving the Euler function and Smarandache function Yi Yuan Research Center for Basic Science, Xi’an Jiaotong University Xi’an, Shaanxi, P.R.China Abstract For any positive integer n, let φ(n) be the Euler function, and S(n) denotes the Smarandache function. The main purpose of this paper is using the elementary methods to study the number of the solutions of the equation φ(n) = S(nk ), where k is any fixed positive integer, and give all solutions for this equation. Keywords Euler function; Smarandache function; Equation; Solutions. §1. Introduction For any positive integer n, let S(n) denotes the Smarandache function, S(n) is defined as the smallest positive integer m such that n|m!. From the definition one can easily deduce that αk αi 1 α2 if n = pα 1 p2 · · · pk is the factorization of n into prime powers, then S(n) = max{S(pi )}, where the maximum is taken over the i’s from 1 to k. Let φ(n) denotes the Euler function. That is, φ(n) denotes the number of all positive integers not exceeding n which are relatively prime to n. It is clear that φ(n) is a multiplicative function. In this paper, we shall study the number of the solutions of the equation φ(n) = S(nk ), where k is any fixed positive integer. About this problem, it is easy to get that n = 1 be a solution of the equation, but now we don’t know whether it have finite solutions or not. Here, we shall use the elementary methods to solve this problem, and give all solutions for this equation for any fixed positive integer k. That is, we shall prove the following conclusions: Theorem 1. The equation φ(n) = S(n2 ) have three solutions, namely n = 1, 24, 50. Theorem 2. The equation φ(n) = S(n3 ) have three solutions, namely n = 1, 48, 98. Theorem 3. The equation φ(n) = S(n4 ) has one solution, namely n = 1. Note. Using the similarly method, we can also deduce that the equation φ(n) = S(nk ) have the finite solutions, where k be any fixed positive integer. §2. Proof of the theorem In this section, we will complete the proof of the theorem. First, we need one simple lemma which is necessary in the proof of Theorems. Lemma If p is prime, then S(pk ) ≤ kp. If k < p, then S(pk ) = kp, where k be any fixed positive integer. 1 This work is supported by the N.S.F(10271093) and P.N.S.F of P.R.China 174 Yi Yuan No. 2 Proof. ( See reference [2] ). Now we shall complete the proofs of Theorems. αk 1 α2 The proof of Theorem 1. Let n = pα 1 p2 · · · pk , then from the definitions of S(n) and φ(n) we have 2α i S(n2 ) = max{S(p2α i )} = S(p ), where p be prime, and φ(n) = pα−1 (p − 1)φ(n1 ), where (n1 , p) = 1. That is, the largest common factor n1 and p is 1. It is clear that n = 1 is a solution of the equation φ(n) = S(n2 ). If n > 1 then we will discuss the problem in the following cases: (i) Let α = 1. If p = 2, then S(22 ) = 4, φ(n) = (2 − 1)φ(n1 ), from S(n2 ) = S(22 ) = φ(n) = φ(n1 ), we get φ(n1 ) = 4, so n1 = 5, then n = 22 × 5. But now S(24 · 52 ) = 10 6= φ(22 × 5), hence the equation has no solution in this case. If p ≥ 3, then from Lemma we have S(p2 ) = 2p, φ(n) = (p − 1)φ(n1 ), note that p † (p − 1)φ(n1 ), hence the equation has no solution in this case. (ii) Let α = 2. If p = 2, then S(24 ) = 6 = 2φ(n1 ), no solution. If p = 3, then S(34 ) = 9 = 3 × 2φ(n1 ), no solution. If p = 5, then S(54 ) = 20 = 5 × 4φ(n1 ), so n1 = 2, hence n = 52 × 2 is a solution. If p ≥ 7, then S(p4 ) = 4p = p(p − 1)φ(n1 ), note that p − 1 > 4, hence no solution. (iii) Let α = 3. If p = 2, then S(26 ) = 8 = 4φ(n1 ), so n1 = 3, hence n = 23 × 3 is a solution. If p = 3, then S(36 ) = 15 = 32 × 2φ(n1 ), no solution. If p = 5, then S(56 ) = 25 = 52 × 4φ(n1 ), no solution. If p = 7, then S(76 ) = 42 = 72 × 6φ(n1 ), no solution. If p > 7, then S(p6 ) = 6p = p(p − 1)φ(n1 ), note that p − 1 > 6, no solution. (iv) Let α = 4. If p = 2, then S(28 ) = 10 = 8φ(n1 ), no solution. If p ≥ 3, from Lemma we have S(p2α ) < 2pα, note that φ(n) = pα−1 (p − 1)φ(n1 ) and pα−1 > 2pα, no solution. (v) Let α = 5. If p = 2, then S(210 ) = 12 = 24 φ(n1 ), no solution. If p ≥ 3, from Lemma we have S(p2α ) < 2pα, note that φ(n) = pα−1 (p − 1)φ(n1 ) and α−1 p > 2pα, no solution. (vi) Let α ≥ 6. If p ≥ 2, from Lemma we have S(p2α ) < 2pα, note that φ(n) = pα−1 (p − 1)φ(n1 ) and pα−1 > 2pα, no solution. Combining (i) to (vi), we may immediately get that the equation φ(n) = S(n2 ) have three solutions, namely n = 1, 24, 50. This completes the proof of Theorem 1. Similarly, using the same methods we can also deduce the results of Theorem 2 and Theorem 3. This complete the proofs of Theorems. Vol. 1 An equation involving the Euler function and Smarandache function 175 For the general positive integer k, from the methods of proving our Theorems we can get that the equation φ(n) = S(nk ) have finite solutions. References [1 ] F.Smarandache, Only problems, not solutions, Chicago, Xiquan Publ. House, 1993. [2] Mark Farris and Patrick Mitchell, Bounding the Smarandache function, Smarandache notions journal, 13(2002). [3] Pan Chengddong and Pan Chengbiao, Elementary number theory. Beijing, Beijing University Press, 1992. [4] Tom M.Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag, 1976. Scientia Magna Vol. 1 (2005), No. 2, 176-178 Recursive Palindromic Smarandache Values Jason Earls R.R. 1-43-05 Fritch, TX 79036 email: jason earls@hotmail.com Abstract In [1] Recursive Prime Numbers were studied and shown to be finite. This article deals with the same ”recursive” topic, but applies the method to numbers whose Smarandache value, S(n), gives a palindromic number. Here, S(n) denotes the Smarandache function of least m such that n divides m!, and a palindrome is an integer that reads the same forwards and backwards (23432, for example). This sequence of recursive palindromic Smarandache values is shown to be finite with 1514384 being the last term. Recursive palindromic Smarandache values (RPSV) are integers n > 0, such that S(n) gives a palindromic value, and repeatedly deleting the rightmost digits of n and taking S(n) at each step also gives a palindromic value until only a single digit remains. (Note that the numbers are not permitted to have zeroes.) Example: n S(n) 94649 1514384 373 151438 797 15143 1514 757 151 151 15 5 1 1 The same algorithm outlined in [1] was used to generate all RPSV sets beginning with each digit 1 through 9. To summarize the basic algorithm, two arrays are defined: A1 , contains only the initial digit, then A2 is filled with any integers that give palindromic Smarandache values after multiplying the integers in A1 by 10 and adding y, with 1 ≤ y ≤ 9. A1 is then updated with the A2 values. This process is repeated until no solutions are found and thus A2 is empty. Vol. 1 Recursive Palindromic Smarandache Values 177 This is enough to prove that the sequence is finite. And using this algorithm, RPSVs were found to be finite with 1514384 being the last term. As in [1], genetic trees can be constructed from each digit for visualization and comparison purposes. Below, only the genetic tree for the digit 5 is produced. It is left to readers so inclined to construct the other trees. (However, the full sequence of RPSV numbers is given at the end of this article.) Tree of recursive palindromic Smarandache values with starting digit 5: 5 54 55 543 567 56 5436 54362 54365 543654 To show that the numbers in the genetic tree above are recursively palindromic when S(n) is applied, let us demonstrate with 54365: n S(n) 54365 131 5436 151 543 181 54 9 5 5 Unsolved Questions: What is the sequence of RPSVs when the leftmost digits are repeatedly deleted? Is the sequence finite? Full Sequence of RPSVs: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16, 18, 21, 22, 24, 27, 28, 32, 33, 35, 36, 42, 44, 45, 48, 54, 55, 56, 63, 64, 66, 72, 77, 81, 84, 88, 96, 99, 112, 121, 126, 128, 144, 151, 154, 162, 165, 168, 181, 189, 216, 224, 242, 275, 288, 324, 336, 352, 353, 362, 363, 448, 453, 484, 543, 567, 648, 724, 726, 727, 847, 968, 1212, 1267, 1441, 1448, 1512, 1514, 1515, 1629, 1812, 1815, 1818, 2424, 2751, 2757, 2882, 3247, 3535, 3537, 3624, 3629, 3635, 178 Jason Earls No. 2 3636, 4536, 4847, 4848, 4849, 5436, 7248, 7272, 7277, 8472, 12127, 12672, 15125, 15143, 18154, 18181, 24245, 27512, 27573, 27576, 32476, 35353, 36359, 36362, 48471, 54362, 54365, 72724, 72727, 72771, 126723, 151436, 151437, 151438, 181542, 181543, 275127, 275762, 363594, 363629, 484718, 543654, 1514384. References [1] S. Tabirca and K. Reynolds, Recursive prime numbers, Smarandache Notions Journal 14(2004), 133-138. Scientia Magna Vol. 1 (2005), No. 2, 179-187 Smarandache Idempotents in finite ring Zn and in Group Ring ZnG W.B.Vasantha and Moon K.Chetry Department of Mathematics I.I.T. Madras, Chennai Abstract In this paper we analyze and study the Smarandache idempotents (S-idempotents) in the ring Zn and in the group ring Zn G of a finite group G over the finite ring Zn . We have shown the existance of Smarandache idempotents (S-idempotents) in the ring Zn when n = 2m p (or 3p), where p is a prime > 2 (or p a prime > 3). Also we have shown the existance of Smarandache idempotents (S-idempotents) in the group ring Z2 G and Z2 Sn where n = 2m p (p a prime of the form 2m t + 1). §1. Introduction This paper has 4 sections. In section 1, we just give the basic definition of S-idempotents in rings. In section 2, we prove the existence of S-idempotents in the ring Zn where n = 2m p, m ∈ N and p is an odd prime. We also prove the existence of S-idempotents for the ring Zn where n is of the form n = 3p, p is a prime greater than 3. In section 3, we prove the existence of S-idempotents in group rings Z2 G of cyclic group G over Z2 where order of G is n, n = 2m p (p a prime of the form 2m t + 1). We also prove the existence of S-idempotents for the group ring Z2 Sn where n = 2m p (p a prime of the form 2m t + 1). In the final section, we propose some interesting number theoretic problems based on our study. Here we just recollect the definition of Smarandache idempotents (S-idempotent) and some basic results to make this paper a self contained one. Definition 1.1[5]. Let R be a ring. An element x ∈ R 0 is said to be a Smarandache idempotent (S-idempotent) of R if x2 = x and there exist a ∈ R x, 0 such that i. ii. Example 1.1. xa = x a2 = x or ax = a. Let Z1 0 = {0, 1, 2, . . . , 9} be the ring of integers modulo 10. Here 62 ≡ 6(mod10), 42 ≡ 6(mod10) and 6 · 4 ≡ 4(mod10). 180 W.B.Vasantha and Moon K.Chetry No. 2 So 6 is a S-idempotent in Z10 . Example 1.2. Take Z12 = {0, 1, 2, . . . , 11} the ring of integers modulo 12. Here 42 ≡ 4(mod12), 82 ≡ 4(mod12) and 4 · 8 ≡ 8(mod12). So 4 is a S-idempotent in Z12 . Example 1.3. In Z30 = {0, 1, 2, . . . , 29} the ring of integers modulo 30, 25 is a Sidempotent. As 252 ≡ 25(mod30), 52 ≡ 25(mod30) and 25 · 5 ≡ 5(mod30). So 25 is a S-idempotent in Z30 . Theorem 1.1 [5]. Let R be a ring. If x ∈ R is a S-idempotent then it is an idempotent in R. Proof. From the very definition of S-idempotents. §2. S-idempotents in the finite ring Zn In this section, we find conditions for Zn to have S-idempotents and prove that when n is of the form 2m p, p a prime ¿2 or n = 3p (p a prime ¿3) has S-idempotents. We also explicitly find all the S-idempotents. Theorem 2.1. Zp = {0, 1, 2, . . . , p − 1}, the prime field of characteristic p, where p is a prime has no non-trivial S-idempotents. Proof. Straightforward, as every S-idempotents are idempotents and Zp has no nontrivial idempotents. Theorem 2.2: The ring Z2p , where p is an odd prime has S-idempotents. Proof. Here p is an odd prime, so p must be of the form 2m + 1 i.e p = 2m + 1. Take x=p+1 and a = p − 1. Here p2 = (2m + 1)2 = 4m2 + 4m + 1 = 2m(2m + 1) + 2m + 1 = 2pm + p ≡ p(mod2p). So p2 ≡ p(mod2p). Vol. 1 Smarandache Idempotents in finite ring Zn 181 Again x2 = (p + 1)2 ≡ p2 + 1(mod2p) ≡ p + 1(mod2p). Therefore x2 = x. Also a2 = (p − 1)2 ≡ p + 1(mod2p), therefore a2 = x. And xa = (p + 1)(p − 1) = p2 − 1 ≡ p − 1(mod2p) therefore xa = a. So x = p + 1 is a S-idempotent in Z2p . Example 2.1. Take Z6 = Z2·3 = {0, 1, 2, 3, 4, 5} the ring of integers modulo 6. Then x = 3 + 1 = 4 is a S-idempotent. As x2 = 42 ≡ 4(mod6), take a = 2, then a2 = 22 ≡ 4(mod6). Therefore a2 = x, and xa = 4 · 2 ≡ 2(mod6) i.e xa = a. Theorem 2.3. The ring Z22 p , p a prime > 2 and is of the form 4m + 1 or 4m + 3 has (at least) two S-idempotents. Proof. Here p is of the form 4m + 1 or 4m + 3. If p = 4m + 1, then p2 ≡ p(mod22 p). As p2 = (4m + 1)2 = 16m2 + 8m + 1 = 4m(4m + 1) + 4m + 1 = 4pm + p ≡ p(mod22 p), 182 W.B.Vasantha and Moon K.Chetry No. 2 therefore p2 ≡ p(mod22 p). Now, take x = 3p + 1 and a = p − 1 then x2 = (3p + 1)2 = 9p2 + 6p + 1 ≡ 9p + 6p + 1(mod22 p) ≡ 3p + 1(mod22 p) therefore a2 = x. And xa = = (3p + 1)(p − 1) 3p2 − 3p + p − 1 ≡ p − 1(mod22 p) therefore xa = a. So x is an S-idempotent. Similarly, we can prove that y = p, (here take a = 3p) is another S-idempotent. These are the only two S-idempotents in Z22 p when p = 4m + 1. If p = 4m + 3, then p2 ≡ 3p(mod22 p). As above, we can show that x = p + 1, (a = 3p − 1) and y = 3p, (a = p) are the two S-idempotents. So we are getting a nice pattern here for S-idempotents in Z22 p : I. If p = 4m + 1, then x = 3p + 1, (a = p − 1) and y = p, (a = 3p) are the two S-idempotents. II. If p = 4m + 3, x = p + 1, (a = 3p − 1) and y = 3p, (a = p) are the two S-idempotents. Example 2.2. Take Z22 ·5 = {0, 1, . . . , 19}, here 5 = 4 · 1 + 1. So x = 3 · 5 + 1 = 16, (a = 5 − 1 = 4) is an S-idempotent. As 162 ≡ 16(mod20), 42 ≡ 16(mod20) and 16 · 4 ≡ 4(mod20). Also y = 5, (a = 3 · 5 = 15) is another S-idempotent. As 52 ≡ 5(mod20), 152 ≡ 5(mod20) and 5 · 15 ≡ 15(mod20). Example 2.3. In the ring Z22 ·7 = {0, 1, . . . , 27}, here 7 = 4 · 1 + 3, x = 7 + 1 = 8, (a = 3 · 7 − 1 = 20) is an S-idempotent. As 82 ≡ 8(mod28), 202 ≡ 8(mod28) and 8 · 20 ≡ 20(mod28). Also y = 3 · 7 = 21, (a = 7) is another S-idempotent. As 212 ≡ 21(mod28), 72 ≡ 21(mod28) and 21 · 7 ≡ 7(mod28). Theorem 2.4. The ring Z23 p , p a prime > 2 has (at least) two S-idempotents of φ(23 ) types (where φ(n) is the number of integer less than n and relatively prime to n). Proof. As p is prime > 2. So p is one of the 8m + 1, 8m + 3, 8m + 5, 8m + 7. Now we will get the following two S-idempotents for each φ(23 ) = 4 types of prime p. I. If p = 8m + 1, then x = 7p + 1, (a = p − 1) and y = p, (a = 7p) are S-idempotents. II. If p = 8m + 3, then x = 5p + 1, (a = 3p − 1) and y = 3p, (a = 5p) are S-idempotents. III. If p = 8m + 5, then x = 3p + 1, (a = 5p − 1) and y = 5p, (a = 3p) are S-idempotents. IV. If p = 8m + 7, then x = p + 1, (a = 7p − 1) and y = 7p, (a = p) are S-idempotents. Vol. 1 Smarandache Idempotents in finite ring Zn 183 Example 2.4. In the ring Z23 ·3 = {0, 1, . . . , 23}, here 3 = 8 · 0 + 3. So x = 5 · 3 + 1 = 16, (a = 3 · 3 − 1 = 8) is an S-idempotent. As 162 ≡ 16(mod24), 82 ≡ 16(mod24) and 16 · 8 ≡ 8(mod24). Also y = 3 · 3 = 9, (a = 5 · 3 = 15) is another S-idempotent. As 92 ≡ 9(mod24), 152 ≡ 9(mod24) and 9 · 15 ≡ 15(mod24). Example 2.5. Take Z23 ·13 = Z104 = {0, 1, . . . , 103}, here 13 = 8·1+5. So x = 3·13+1 = 40, (a = 5 · 13 − 1 = 64) is an S-idempotent. As 402 ≡ 40(mod104), 642 ≡ 40(mod104) and 40 · 64 ≡ 64(mod104). Also y = 5 · 13 = 65, (a = 3 · 13 = 39) is another S-idempotent. As 652 ≡ 65(mod104), 392 ≡ 65(mod104) and 65 · 39 ≡ 39(mod104). Theorem 2.5. The ring Z24 p , p a prime > 2 has (at least) two S-idempotents for each of φ(24 ) types of prime p. Proof. As above, we can list the S-idempotents for all φ(24 ) = 8 types of prime p. I. If p = 16m + 1, then x = 15p + 1, (a = p − 1) and y = p, (a = 15p) are S-idempotents. II. If p = 16m + 3, then x = 13p + 1, (a = 3p − 1) and y = 3p, (a = 13p) are S-idempotents. III. If p = 16m + 5, then x = 11p + 1, (a = 5p − 1) and y = 5p, (a = 11p) are S-idempotents. IV. If p = 16m + 7, then x = 9p + 1, (a = 7p − 1) and y = 7p, (a = 9p) are S-idempotents. V. If p = 16m + 9, then x = 7p + 1, (a = 9p − 1) and y = 9p, (a = 7p) are S-idempotents. VI. If p = 16m+11, then x = 5p+1, (a = 11p−1) and y = 11p, (a = 5p) are S-idempotents. VII. If p = 16m + 13, then x = 3p + 1, (a = 13p − 1) and y = 13p, (a = 13p) are S-idempotents. VIII. If p = 16m+15, then x = p+1, (a = 15p−1) and y = 15p, (a = p) are S-idempotents. Example 2.6. In the ring Z24 ·17 = Z272 = {0, 1, . . . , 271}, here 17 = 16 · 1 + 1. So x = 15 · 17 + 1 = 256, (a = 17 − 1 = 16) is an S-idempotent. As 2562 ≡ 256(mod272), 162 ≡ 256(mod272) and 256 · 16 ≡ 16(mod272). Also y = 17, (a = 15 · 17 = 255) is another Sidempotent. As 172 ≡ 17(mod272), 2552 ≡ 17(mod272) and 17 · 255 ≡ 255(mod272). We can generalize the above result as followings: Theorem 2.6. The ring Z2n p , p a prime > 2 has (at least) two S-idempotents for each of φ(2n ) types of prime p. Proof. Here p is one of the φ(2n ) form: 2n m1 + 1, 2n m2 + 3, ... 2n mφ(2n ) + (2n − 1). We can find the two S-idempotents for each p as above. We are showing here for the prime p = 2n m1 + 1 only. If p = 2n m1 + 1, then x = (2n − 1)p + 1, (a = p − 1) and y = p, (a = (2n − 1)p) are S-idempotents. Similarly we can find S-idempotents for each of the φ(2n ) form of prime p. Theorem 2.7. The ring Z3p , p a prime > 3 has (at least) two S-idempotents of φ(3) types. 184 W.B.Vasantha and Moon K.Chetry No. 2 Proof. Here p can be one of the form 3m + 1 or 3m + 2. We can apply the Theorem 2.6 for Z3p also. I. If p = 3m + 1, then x = 2p + 1, (a = p − 1) and y = p, (a = 2p) are S-idempotents. II. If p = 3m + 2, then x = p + 1, (a = 2p − 1) and y = 2p, (a = p) are S-idempotents. Example 2.7. In the ring Z3·5 = Z15 = {0, 1, . . . , 14}, here 5 = 3 · 1 + 2. So x = 5 + 1 = 6, (a = 2 · 5 − 1 = 9) is an S-idempotent. As 62 ≡ 6(mod15), 92 ≡ 6(mod15) and 6 · 9 ≡ 9(mod15). Also y = 2 · 5 = 10, (a = 5) is another S-idempotent. As 102 ≡ 10(mod15), 52 ≡ 10(mod15) and 10 · 5 ≡ 5(mod15). Remark: The above result is not true for the ring Z32 p , p prime > 3. As, for p = 9m + 5; x = 4p + 1, (a = 5p − 1) should be an S-idempotent from the above result. But we see it is not the case in general; for take the ring Z32 ·23 = Z207 = {0, 1, . . . , 206}. Here p = 9 · 2 + 5. Now take x = 4 · 23 + 1 = 93 and a = 5 · 23 − 1 = 114. But x2 6≡ x(mod207). So x is not even an idempotent. So x = 4p + 1 is not an S-idempotent of Z32 p . §3. S-idempotents in the group rings Z2 G Here we prove the existance of Smarandache idempotents for the group rings Z32 p of the cyclic group G of order 2n p where p is a prime of the form 2n t + 1. Example 3.2. Let G = {g/g 52 = 1} be the cyclic group of order 22 · 13. Consider the group ring Z2 G of the group G over Z2 . Take x = 1 + g 4 + g 8 + g 12 + . . . + g 44 + g 48 and a = 1 + g 2 + g 4 + . . . + g 22 + g 24 then x2 = x, and a2 = x also x · a = x. So x = 1 + g 4 + g 8 + g 12 + . . . + g 44 + g 48 is a S-idempotent in Z2 G. Theorem 3.1. Let Z2 G be the group ring of the finite cyclic group G of order 22 p, where p is a prime of the form 22 m + 1, then the group ring Z2 G has non-trivial S-idempotents. Proof. Here G is a cyclic group of order 22 p, where p of the form 22 m + 1. Take x = 1 + g 4 + g 8 + . . . + g 16m and a = 1 + g 2 + g 4 + . . . + g 8m Vol. 1 Smarandache Idempotents in finite ring Zn 185 then x2 = (1 + g 4 + g 8 + . . . + g 16m )2 = 1 + g 4 + g 8 + . . . + g 16m = x. And a2 = (1 + g 2 + g 4 + . . . + g 8m )2 = 1 + (g 2 )2 + (g 4 )2 + . . . + (g 8m )2 = x. Also x · a = (1 + g 4 + g 8 + . . . + g 16m )(1 + g 2 + g 4 + . . . + g 8m ) = 1 + g 4 + g 8 + . . . + g 16m = x. So x = 1 + g 4 + g 8 + . . . + g 16m is a S-idempotent in Z2 G. Example 3.3. Let G = {g/g 136 = 1} be the cyclic group of order 23 · 17. Consider the group ring Z2 G of the group G over Z2 . Take x = 1 + g 8 + g 16 + . . . + g 128 and a = 1 + g 4 + g 8 + . . . + g 64 then x2 = (1 + g 8 + g 16 + . . . + g 128 )2 = 1 + g 8 + g 16 + . . . + g 128 = x. And a2 = (1 + g 4 + g 8 + . . . + g 64 )2 = 1 + (g 4 )2 + (g 8 )2 + . . . + (g 64 )2 = x. Also x · a = (1 + g 8 + g 16 + . . . + g 128 )(1 + g 4 + g 8 + . . . + g 64 ) = 1 + g 8 + g 64 + . . . + g 128 = x. So x = 1 + g 8 + g 16 + . . . + g 128 is a S-idempotent in Z2 G. 186 W.B.Vasantha and Moon K.Chetry No. 2 Theorem 3.2. Let Z2 G be the group ring of a finite cyclic group G of order 23 p, where p is a prime of the form 23 m + 1, then the group ring Z2 G has non-trivial S-idempotents. Proof. Here G is a cyclic group of order 23 p, where p of the form 23 m + 1. Take x = 1 + g 8 + g 16 + . . . + g 8(p−1) and a = 1 + g 4 + g 8 + . . . + g 4(p−1) then x2 = (1 + g 8 + g 16 + . . . + g 8(p−1) )2 = 1 + g 8 + g 16 + . . . + g 8(p−1) = x. And a2 = (1 + g 4 + g 8 + . . . + g 4(p−1) )2 = 1 + (g 4 )2 + (g 8 )2 + . . . + (g 8(p−1) )2 = x. Also x · a = (1 + g 8 + g 16 + . . . + g 8(p−1) )(1 + g 4 + g 8 + . . . + g 4(p−1) ) = 1 + g 8 + g 16 + . . . + g 8(p−1) = x. So x = 1 + g 8 + g 16 + . . . + g 8(p−1) is a S-idempotent in Z2 G. We can generalize the above two results as followings: Theorem 3.3. Let Z2 G be the group ring of a finite cyclic group G of order 2n p, where p is a prime of the form 2n t + 1, then the group ring Z2 G has non-trivial S-idempotents. Proof. Here G is a cyclic group of order 2n p, where p of the form 2n t + 1. Take n n n x = 1 + g 2 + g 2 ·2 + . . . + g 2 (p−1) and a = 1 + g2 n−1 + g2 n−1 ·2 + . . . + g2 n−1 ·(p−1) then x2 n = (1 + g 2 + g 2 = 1+g 2 n +g n ·2 n 2 ·2 + . . . + g2 + ... + g n (p−1) 2 ) n 2 (p−1) = x. And a2 = (1 + g 2 n−1 2 n−1 = 1 + (g = x. + g2 2 n−1 ) + (g 2 ·2 + . . . + g2 n−1 ·2 2 n−1 ·(p−1) 2 ) + . . . + (g ) 2n−1 ·(p−1) 2 ) Vol. 1 187 Smarandache Idempotents in finite ring Zn Also x·a = = n (1 + g 2 + g 2 1+g 2 n +g n n ·2 2 ·2 + . . . + g2 + ... + g n (p−1) )(1 + g 2 n−1 + g2 n−1 ·2 + . . . + g2 n−1 ·(p−1) ) n 2 (p−1) = x. n n n So x = 1 + g 2 + g 2 ·2 + . . . + g 2 (p−1) is a S-idempotent in Z2 G. Corollary 3.1. Let Z2 Sn be the group ring of a symmetric group Sn where n = 2n p, and p is a prime of the form 2n t + 1, then the group ring Z2 Sn has non-trivial S-idempotents. Proof. Here Z2 Sn is a group ring where n = 2n p, and p of the form 2n t+1. Clearly Z2 Sn contains a finite cyclic group of order 2n p. Then by the Theorem 3.3, Z2 Sn has a non-trivial S-idempotent. §4. Conclusions Here we have mainly proved the existance of S-idempotents in certain types of group rings. But it is interesting to enumerate the number of S-idempotents for the group rings Z2 G and Z2 Sn in the Theorem 3.3 and Corollary 3.1. We feel that Z2 G can have only one S-idempotent but we are not in a position to give a proof for it. Also, the problem of finding S-idempotents in Zp Sn (and Zp G) where (p, n) = 1 (and (p, |G|) = 1) or (p, n) = d 6= 1 (and (p, |G|) = d 6= 1) are still interesting number theoretic problems. References [1] I.G.Connel, On the group ring, Can.J.Math. 15(1963), 650-685. [2] Kim E.T., Idempotents in some group rings, Bull Korean math.soc., 2(1987), 77-81. [3] C.P.Milies and S.K.Sehgal, An Introduction to group rings, algebras and applications, Kluwer Academic Publishers, Dordrecht, Netherlands,(2002). [4] D.S.Passman, The algebraic structure of group rings, Wiley interscience (1997). [5] W.B.Vasantha Kandasamy, Smarandache rings, American research press, Rehoboth (2002). Scientia Magna Vol. 1 (2005), No. 2, 188-195 Smarandache Idempotents in Loop Rings ZtLn(m) of the Loops Ln(m) W.B.Vasantha and Moon K. Chetry Department of Mathematics, I.I.T.Madras, Chennai Abstract In this paper we establish the existence of S-idempotents in case of loop rings Zt Ln (m) for a special class of loops Ln (m); over the ring of modulo integers Zt for a specific value of t. These loops satisfy the conditions gi2 for every gi ∈ Ln (m). We prove Zt Ln (m) has an S-idempotent when t is a perfect number or when t is of the form 2i p or 3i p (where p is an odd prime) or in general when t = pi1 p2 (p1 and p2 are distinct odd primes), It is important to note that we are able to prove only the existence of a single S-idempotent; however we leave it as an open problem whether such loop rings have more than one S-idempotent. §1. Basic Results This paper has three sections. In section one, we give the basic notions about the loops Ln (m) and recall the definition of S-idempotents in rings. In section two, we establish the existence of S-idempotents in the loop ring Zt Ln (m). In the final section, we suggest some interesting problems based on our study. Here we just give the basic notions about the loops Ln (m) and the definition of Sidempotents in rings. Definition 1.1 [4]. Let R be a ring. An element x ∈ R \ {0} is said to be a Smarandache idempotents (S-idempotent) of R if x2 = x and there exist a ∈ R \ {x, 0} such that i. a2 = x ii. xa = x or ax = a. For more about S-idempotent please refer [4]. Definition 1.2 [2]. A positive integer n is said to be a perfect number if n is equal to the sum of all its positive divisors, excluding n itself. e.g. 6 is a perfect number. As 6 = 1 + 2 + 3. Definition 1.3 [1]. A non-empty set L is said to form a loop, if in L is defined a binary operation, called product and denoted by 0 .0 such that 1. For a, b ∈ L we have a.b ∈ L. (closure property.) 2. There exists an element e ∈ L such that a.e = e.a = a for all a ∈ L. (e is called the identity element of L.) 3. For every ordered pair (a, b) ∈ L × L there exists a unique pair (x, y) ∈ L × L such that ax = b and ya = b. Vol. 1 Smarandache Idempotents in Loop Rings Zt Ln (m) of the Loops Ln (m) 189 Definition 1.4 [3]. Let Ln (m) = {e, 1, 2, 3, · · · , n} be a set where n > 3, n is odd and m is a positive integer such that (m, n) = 1 and (m − 1, n) = 1 with m < n. Define on Ln (m), a binary operation 0 .0 as following: i. e.i = i.e = i for all i ∈ Ln (m) \ {e} ii. i2 . = e for all i ∈ Ln (m) iii. i.j = t, where t ≡ (mj − (m − 1)i)(modn) for all i, j ∈ Ln (m), i 6= e and j 6= e. Then Ln (m) is a loop. This loop is always of even order; further for varying m, we get a class of loops of order n + 1 which we denote by Ln = {Ln (m)|n > 3, n is odd and (m, n) = 1, (m − 1, n) = 1 with m < n}. Example 1.1 [3]. Consider L5 (2) = {e, 1, 2, 3, 4, 5}. The composition table for L5 (2) is given below: · e 1 2 3 4 5 e e 1 2 3 4 5 1 1 e 3 5 2 4 2 2 5 e 4 1 3 3 3 4 1 e 5 2 4 4 3 5 2 e 1 5 5 2 4 1 3 e This loop is non-commutative and non-associative and of order 6. §2. Existence of S-idempotents in the Loop Rings Zt Ln (m) In this section we will prove the existence of an S-idempotent in the loop ring Zt Ln (m) when t is an even perfect number. Also we will prove that the loop ring Zt Ln (m) has an S-idempotent when t is of the form 2i p or 3i p (where p is an odd prime) or in general when t = pi1 p2 (p1 and p2 are distinct odd primes). Theorem 2.1. Let Zt Ln (m) be a loop ring, where t is an even perfect number of the form t = 2s (2s+1 − 1) for some s > 1, then α = 2s + 2s gi ∈ Zt Ln (m) is an S-idempotent. Proof. As t is an even perfect number, t must be of the form t = 2s (2s+1 − 1), for some s > 1 where 2s+1 − 1 is a prime. Consider α = 2s + 2s gi ∈ Zt Ln (m). Choose β = (t − 2s ) + (t − 2s )gi ∈ Zt Ln (m). 190 W.B.Vasantha and Moon K. Chetry No. 2 Clearly α2 = (2s + 2s gi )2 = 2.22s (1 + gi ) ≡ 2s (1 + gi ) = α. [2s .2s+1 ≡ 2s (mod t)] Now β2 = ((t − 2s ) + (t − 2s )gi )2 = 2.(t − 2s )2 (1 + gi ) ≡ 2s (1 + gi ) = α. Also αβ = [2s + 2s gi ][(t − 2s ) + (t − 2s )gi ] = 2s (1 + gi )(t − 2s )(1 + gi ) ≡ −2.2s .2s (1 + gi ) ≡ (t − 2s )(1 + gi ) = β. So we get α2 = α, β 2 = α and αβ = β. Therefore α = 2s + 2s gi is an S-idempotent. Example 2.1. Take the loop ring Z6 Ln (m). Here 6 is an even perfect number. As 6 = 2.(2s − 1), so α = 2 + 2gi is an S-idempotent. For α2 = (2 + 2gi )2 ≡ 2 + 2gi = α. Choose now β = (6 − 2) + (6 − 2)gi . then β2 = (4 + 4gi )2 ≡ (2 + 2gi ) = α. And αβ = (2 + 2gi )(4 + 4gi ) = 8 + 8gi + 8gi + 8 ≡ 4 + 4gi = β. Vol. 1 Smarandache Idempotents in Loop Rings Zt Ln (m) of the Loops Ln (m) 191 So α = 2 + 2gi is an S-idempotent. Theorem 2.2. Let Z2p Ln (m) be a loop ring where p is an odd prime such that p | 2t0 +1 −1 for some t0 ≥ 1, then α = 2t0 + 2t0 gi ∈ Z2p Ln (m) is an S-idempotent. Proof. Suppose p | 2t0 +1 − 1 for some t0 ≥ 1. Take α = 2t0 + 2t0 gi ∈ Z2p Ln (m) and β = (2p − 2t0 ) + (2p − 2t0 )gi ∈ Z2p Ln (m). Clearly α2 = (2t0 + 2t0 gi )2 = 2.22t0 (1 + gi ) = 2t0 +1 .2t0 (1 + gi ) ≡ 2t0 (1 + gi ) = α. As 2t0 .2t0 +1 ≡ 2t0 (mod 2p) Since 2t0 +1 ≡ 1(mod p) ⇔ 2t0 .2t0 +1 ≡ 2t0 (mod 2p) for gcd(2t0 , 2p) = 2, t0 ≥ 1. Also β2 = [(2p − 2t0 ) + (2p − 2t0 )gi ]2 = 2(2p − 2t0 )2 (1 + gi ) ≡ 2.22t0 (1 + gi ) = 2t0 +1 .2t0 (1 + gi ) ≡ 2t0 (1 + gi ) = α. And αβ = [2t0 + 2t0 gi ][(2p − 2t0 ) + (2p − 2t0 )gi ] ≡ −2t0 (1 + gi )2t0 (1 + gi ) = −2.2t0 (1 + gi ) ≡ (2p − 2t0 )(1 + gi ) = β. So we get α2 = α, β 2 = α and αβ = β. Hence α = 2t0 + 2t0 gi is an S-idempotent. Example 2.2. Consider the loop ring Z10 Ln (m). Here 5 | 23+1 − 1, so t0 = 3. Take 192 W.B.Vasantha and Moon K. Chetry No. 2 α = 23 + 23 gi and β = 2 + 2gi . Now α2 = (8 + 8gi )2 = 64 + 128gi + 64 ≡ 8 + 8gi = α. And β2 = (2 + 2gi )2 = 4 + 8gi + 4 ≡ 8 + 8gi = α. Also αβ = (8 + 8gi )(2 + 2gi ) = 16 + 16gi + 16gi + 16 ≡ 2 + 2gi = β. So α = 8 + 8gi is an S-idempotent. Theorem 2.3. Let Z2i p Ln (m) be a loop ring where p is an odd prime such that p | 2t0 +1 −1 for some t0 ≥ i, then α = 2t0 + 2t0 gi ∈ Z2i p Ln (m) is an S-idempotent. Proof. Note that p | 2t0 +1 − 1 for some t0 ≥ i. Therefore 2t0 +1 ≡ 1(mod p) for some t0 ≥ i ⇔ 2t0 .2t0 +1 ≡ 2t0 (mod 2i p) as gcd(2t0 , 2i p) = 2i , t0 ≥ 1. Now take α = 2t0 + 2t0 gi ∈ Z2i p Ln (m) and β = (2i p − 2t0 ) + (2i p − 2t0 )gi ∈ Z2i p Ln (m). Then it is easy to see that α2 = α, β 2 = α and αβ = β. Hence α = 2t0 + 2t0 gi is an S-idempotent. Example 2.3. Take the loop ring Z23 .7 Ln (m). Here 7 | 25+1 − 1, so t0 = 5. Take α = 25 + 25 gi and β = (23 .7 − 25 ) + (23 .7 − 25 )gi . Vol. 1 Smarandache Idempotents in Loop Rings Zt Ln (m) of the Loops Ln (m) 193 Now α2 = (32 + 32gi )2 = 1024 + 2048gi + 1024 ≡ 32 + 32gi = α. And β2 = (24 + 24gi )2 = 576 + 1152gi + 576 ≡ 24 + 24gi = α. Also αβ = (32 + 32gi )(24 + 24gi ) ≡ 24 + 24gi = β. So α = 32 + 32gi is an S-idempotent. Theorem 2.4. Let Z3i p Ln (m) be a loop ring where p is an odd prime such that p | 2.3t0 −1 for some t0 ≥ i, then α = 3t0 + 3t0 gi ∈ Z3i p Ln (m) is an S-idempotent. Proof. Suppose p | 2.3t0 − 1 for some t0 ≥ i. Take α = 3t0 + 3t0 gi ∈ Z3i p Ln (m) and β = (3i p − 3t0 ) + (3i p − 3t0 )gi ∈ Z3i p Ln (m). Then α2 = (3t0 + 3t0 gi )2 = 2.32t0 (1 + gi ) = 2.3t0 3t0 (1 + gi ) ≡ 3t0 (1 + gi ) = α. As 2.3t0 ≡ 1(mod p) for some t0 ≥ i ⇔ 2.3t0 .3t0 ≡ 3t0 (mod 3i p) as gcd(3t0 , 3i p) = 3i , t0 ≥ 1. Similarly β2 = α So α = 3t0 + 3t0 gi is an S-idempotent. and αβ = β. 194 W.B.Vasantha and Moon K. Chetry No. 2 Example 2.4. Take the loop ring Z32 .5 Ln (m). Here 5 | 2.35 − 1, so t0 = 5. Take α = 35 + 35 gi and β = (32 .5 − 35 ) + (32 .5 − 35 )gi . Now α2 = (18 + 18gi )2 ≡ 18 + 18gi = α. And β2 = (27 + 27gi )2 ≡ 18 + 18gi = α. Also αβ = β. So α = 35 + 35 gi is an S-idempotent. We can generalize Theorem 2.3 and Theorem 2.4 as following: Theorem 2.5. Let Zpi1 p2 Ln (m) be a loop ring where p1 and p2 are distinct odd primes and p2 | 2.pt10 − 1 for some t0 ≥ i, then α = pt10 + pt10 gi ∈ Zpi1 p2 Ln (m) is an S-idempotent. Proof. Suppose p2 | 2.pt10 − 1 for some t0 ≥ i. Take α = pt10 + pt10 gi ∈ Zpi1 p2 Ln (m) and β = (pi1 p2 − pt10 ) + (pi1 p2 − pt10 )gi ∈ Zpi1 p2 Ln (m). Then α2 = (pt10 + pt10 gi )2 0 = 2.p2t 1 (1 + gi ) = 2.pt10 pt10 (1 + gi ) ≡ pt10 (1 + gi ) = α. As 2.pt10 ≡ 1(mod p2 ) for some t0 ≥ i . ⇔ 2.pt10 .pt10 ≡ pt10 (mod pi1 p2 ) as gcd(pt10 , pi1 p2 ) = pi1 , t0 ≥ i. Similarly β2 = α So α = pt10 + pt10 gi is an S-idempotent. and αβ = β. Vol. 1 Smarandache Idempotents in Loop Rings Zt Ln (m) of the Loops Ln (m) 195 §3. Conclusion We see in all the 5 cases described in the Theorem 2.1 to 2.5 we are able to establish the existence of one non-trivial S-idempotent. however we are not able to prove the uniqueness of this S-idempotent. Hence we suggest the following problems: • Does the loop rings described in the Theorems 2.1 to 2.5 can have more than one Sidempotent? • Does the loop rings Zt Ln (m) have S-idempotent when t is of the form t = p1 p2 . . . ps where p1 p2 . . . ps are distinct odd primes? References [1] [2] [3] [4] 2002. Bruck R.H, A survey of binary system, Spring Verlag, 1958. Burton David, Elementary Number Theory, Universal Book Stall. New Delhi, 1998. Singh S.V., On a new class of loops and loop rings. PhD thesis, IIT Madras, 1994. Vasantha Kandasamy, W.B. Smarandache Rings. American Reseach Press, Rehoboth, SCIENTIA MAGNA An international journal