OFFSET
1,1
COMMENTS
People who search for repunit primes or repdigit primes may be looking for this entry.
The indices of primes with digital product (i.e., product of digits) equal to 1.
As of August 2014, only the first five repunits, through (10^1031-1)/9, have been proved prime. The last four repunits are known only to be probable primes and have not been proved to be prime. - Robert Baillie, Aug 17 2014
These indices p must also be prime. If p is not prime, say p = m*n, then 10^(m*n) - 1 = ((10^m)^n) - 1 => 10^m - 1 divides 10^(m*n) - 1. Since 9 divides 10^m - 1 or (10^m - 1)/9 = q, it follows q divides (10^p - 1)/9. This is a result of the identity, a^n - b^n = (a - b)(a^(n-1) + a^(n-2)*b + ... + b^(n-1)). - Cino Hilliard, Dec 23 2008
The numbers R_n = 11...111 = (10^n - 1)/9 with n in this sequence A004023, except for n = 2, are prime repunits in base ten, so they are prime Brazilian numbers belonging to A085104. [See Links: Les nombres brésiliens.] - Bernard Schott, Dec 24 2012
Search limit is 10800000, currently. - Serge Batalov, Jul 01 2021
REFERENCES
J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 19, pp 6, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, Section A3.
Graham, Knuth and Patashnik, Concrete Mathematics, Addison-Wesley, 1994; see p 146 problem 22.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 142857 at pp. 197-198.
LINKS
Paul Bourdelais, A Generalized Repunit Conjecture, NMBRTHRY, 25 Jun 2009.
John Brillhart, Letter to N. J. A. Sloane, Aug 08 1970
John Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Chris K. Caldwell, The Prime Pages, Top 20: Repunit (lists certified primes with n >= 1000)
Patrick De Geest, Circular Primes
Giovanni Di Maria, Repunit Primes Project
Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
Harvey Dubner, New probable prime repunit, R(49081), Number Theory List, Sep 09 1999.
Harvey Dubner, Repunit R49081 is a probable prime, Math. Comp., 71 (2001), 833-835.
Harvey Dubner, Posting to Number Theory List : Apr 03 2007
Martianus Frederic Ezerman, Bertrand Meyer, and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012-2014.
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See p. 18.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Henri Lifchitz, Mersenne and Fermat primes field
T. Muller, Ist die Folge der Primzahl-quersummen beschrankt?, Elem. Math. 66 (2011) 146-154; doi:10.4171/EM/183.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
Andy Steward, Prime Generalized Repunits
Sam Wagstaff, Jr., The Cunningham Project
E. Wegrzynowski, Nombres 1_[n] premiers
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Repunit Prime
H. C. Williams and Harvey Dubner, The primality of R1031, Math. Comp., 47(176), Oct 1986, 703-711.
EXAMPLE
2 appears because the 2-digit repunit 11 is prime.
3 does not appear because 111 = 3 * 37 is not prime.
19 appears because the 19-digit repunit 1111111111111111111 is prime.
MATHEMATICA
Select[Range[271000], PrimeQ[FromDigits[PadRight[{}, #, 1]]] &] (* Harvey P. Dale, Nov 05 2011 *)
repUnsUpTo[k_] := ParallelMap[If[PrimeQ[#] && PrimeQ[(10^# - 1)/9], #, Nothing] &, Range[k]]; repUnsUpTo[5000] (* Mikk Heidemaa, Apr 24 2017 *)
PROG
(PARI) forprime(x=2, 20000, if(ispseudoprime((10^x-1)/9), print1(x", "))) \\ Cino Hilliard, Dec 23 2008
(Magma) [p: p in PrimesUpTo(500) | IsPrime((10^p - 1) div 9)]; // Vincenzo Librandi, Nov 06 2014
(Python) from sympy import isprime; {print(n, end = ', ') for n in range(1, 10**7) if isprime(n) and isprime(10**n//9)} # (Note that sympy.isprime is only a pseudo-primality test.) - Ya-Ping Lu, Dec 20 2021, edited by M. F. Hasler, Mar 28 2022
CROSSREFS
KEYWORD
hard,nonn,nice,more,changed
AUTHOR
EXTENSIONS
a(6) = 49081 PRP found by Harvey Dubner - posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) Sep 09, 1999; proved prime by Paul Underwood, Mar 21 2022.
a(7) = 86453 found using pfgw (a faster version of PrimeForm) on Oct 26 2000 by Lew Baxter (posting to Number Theory List), Oct 26, 2000; proved prime by Andreas Enge, May 16 2023.
a(8) = 109297 was apparently discovered independently by (in alphabetical order) Paul Bourdelais and Harvey Dubner around Mar 26-28 2007.
a(9) = 270343, was found Jul 11 2007 by Maksym Voznyy and Anton Budnyy, subsequently confirmed as a(9) (see Repunit Primes Project link) by Robert Price, Dec 14 2010
a(10) = 5794777 was found Apr 20 2021 by Ryan Propper and Serge Batalov
a(11) = 8177207 was found May 08 2021 by Ryan Propper and Serge Batalov
STATUS
approved