| Displaying 1-9 of 9 results found.
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Numbers k such that 2^k-1 has only one primitive prime factor.
+10
9
2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208
COMMENTS
Also, numbers k such that A086251(k) = 1. Also, numbers k such that A064078(k) is a prime power. The corresponding primitive primes are listed in A161509. The binary expansion of 1/p has period k and this is the only prime with such a period. The binary analog of A007498. This sequence has many terms in common with A072226. A072226 has the additional term 6; but it does not have terms 18, 20, 21, 54, 147, 342, 602, and 889 (less than 10000).
MATHEMATICA
Select[Range[1000], PrimePowerQ[Cyclotomic[ #, 2]/GCD[Cyclotomic[ #, 2], # ]]&]
The unique primitive prime factor of 2^n-1 for the n in A161508.
+10
5
3, 7, 5, 31, 127, 17, 73, 11, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 241, 2731, 262657, 331, 2147483647, 65537, 599479, 43691, 174763, 61681, 5419, 2796203, 4432676798593, 87211, 15790321, 2305843009213693951, 715827883
COMMENTS
For these primes p, the binary expansion of 1/p has a unique period length. The binary analog of A007615.
MATHEMATICA
Reap[Do[c=Cyclotomic[n, 2]; q=c/GCD[c, n]; If[PrimePowerQ[q], Sow[FactorInteger[q][[1, 1]]]], {n, 100}]][[2, 1]]
Bases in which 3 is a unique-period prime.
+10
5
2, 4, 5, 8, 10, 11, 17, 23, 26, 28, 35, 47, 53, 71, 80, 82, 95, 107, 143, 161, 191, 215, 242, 244, 287, 323, 383, 431, 485, 575, 647, 728, 730, 767, 863, 971, 1151, 1295, 1457, 1535, 1727, 1943, 2186, 2188, 2303, 2591, 2915, 3071, 3455, 3887
COMMENTS
A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 3^t + 1, t >= 1; (b) b = 2^s*3^t - 1, s >= 0, t >= 1.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 3, there are no nontrivial bases, since ord(3,b) <= 2.
EXAMPLE
If b = 3^t + 1, t >= 1, then b - 1 only has prime factor 3, so 3 is a unique-period prime in base b.
If b = 2^s*3^t - 1, t >= 1, then the prime factors of b^2 - 1 are 3 and prime factors of b - 1 = 2^s*3^t - 2, 3 is the only new prime factor so 3 is a unique-period prime in base b.
PROG
(PARI)
p = 3;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));
CROSSREFS
Cf. A040017 (unique primes in base 10), A144755 (unique primes in base 2).
Bases in which 5 is a unique-period prime.
+10
5
2, 3, 4, 6, 7, 9, 19, 24, 26, 39, 49, 79, 99, 124, 126, 159, 199, 249, 319, 399, 499, 624, 626, 639, 799, 999, 1249, 1279, 1599, 1999, 2499, 2559, 3124, 3126, 3199, 3999, 4999, 5119, 6249, 6399, 7999, 9999, 10239, 12499, 12799, 15624, 15626, 15999, 19999, 20479
COMMENTS
A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 5^t + 1, t >= 1; (b) b = 2^s*5^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 7.
For every odd prime p, p is a a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 5, the nontrivial bases are 2, 3, 7.
EXAMPLE
1/5 has period length 4 in base 2. Note that 3 and 5 are the only prime factors of 2^4 - 1 = 15, but 1/3 has period length 2, so 5 is a unique-period prime in base 2.
1/5 has period length 4 in base 3. Note that 2 and 5 are the only prime factors of 3^4 - 1 = 80, but 1/2 has period length 1, so 5 is a unique-period prime in base 3.
1/5 has period length 4 in base 7. Note that 2, 3 and 5 are the only prime factors of 7^4 - 1 = 2400, but 1/2 and 1/3 both have period length 1, so 5 is a unique-period prime in base 7.
PROG
(PARI)
p = 5;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));
Bases in which 7 is a unique-period prime.
+10
5
2, 3, 4, 5, 6, 8, 13, 18, 19, 27, 48, 50, 55, 97, 111, 195, 223, 342, 344, 391, 447, 685, 783, 895, 1371, 1567, 1791, 2400, 2402, 2743, 3135, 3583, 4801, 5487, 6271, 7167, 9603, 10975, 12543, 14335, 16806, 16808, 19207, 21951, 25087, 28671, 33613, 38415, 43903, 50175
COMMENTS
A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 7^t + 1, t >= 1; (b) b = 2^s*7^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 4, 5, 18, 19.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 7, the nontrivial bases are 2, 3, 4, 5, 18, 19.
EXAMPLE
1/7 has period length 3 in base 2. Note that 7 is the only prime factor of 2^3 - 1 = 7, so 7 is a unique-period prime in base 2.
1/7 has period length 3 in base 4. Note that 3, 7 are the only prime factors of 4^3 - 1 = 63, but 1/3 has period length 1, so 7 is a unique-period prime in base 4.
1/7 has period length 3 in base 18. Note that 7, 17 are the only prime factors of 18^3 - 1 = 5831, but 1/17 has period length 1, so 7 is a unique-period prime in base 18.
(1/7 has period length 6 in base 3, 5, 19. Similar demonstrations can be found.)
PROG
(PARI)
p = 7;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));
CROSSREFS
Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).
Bases in which 11 is a unique-period prime.
+10
5
2, 3, 10, 12, 21, 43, 87, 120, 122, 175, 241, 351, 483, 703, 967, 1330, 1332, 1407, 1935, 2661, 2815, 3871, 5323, 5631, 7743, 10647, 11263, 14640, 14642, 15487, 21295, 22527, 29281, 30975, 42591, 45055, 58563, 61951, 85183, 90111, 117127, 123903, 161050, 161052, 170367, 180223, 234255, 247807, 322101, 340735
COMMENTS
A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 11^t + 1, t >= 1; (b) b = 2^s*11^t - 1, s >= 0, t >= 1; (c) b = 2, 3.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 11, the nontrivial bases are 2, 3.
EXAMPLE
1/11 has period length 10 in base 2. Note that 3, 11, 31 are the only prime factors of 2^10 - 1 = 1023, but 1/3 has period length 2 and 1/31 has period length 5, so 11 is a unique-period prime in base 2.
1/11 has period length 5 in base 3. Note that 2, 11 are the only prime factors of 3^5 - 1 = 242, but 1/2 has period length 1, so 11 is a unique-period prime in base 3.
PROG
(PARI)
p = 11;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));
CROSSREFS
Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).
Bases in which 13 is a unique-period prime.
+10
5
2, 3, 4, 5, 12, 14, 22, 23, 25, 51, 103, 168, 170, 207, 239, 337, 415, 675, 831, 1351, 1663, 2196, 2198, 2703, 3327, 4393, 5407, 6655, 8787, 10815, 13311, 17575, 21631, 26623, 28560, 28562, 35151, 43263, 53247, 57121, 70303, 86527, 106495, 114243, 140607, 173055, 212991, 228487, 281215, 346111
COMMENTS
A prime p is called a unique-period prime in base b if there is no other prime q such that the period of the base-b expansion of its reciprocal, 1/p, is equal to the period of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 13^t + 1, t >= 1; (b) b = 2^s*13^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 4, 5, 22, 23, 239.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 13, the nontrivial bases are 2, 3, 4, 5, 22, 23, 239.
EXAMPLE
1/13 has period 12 in base 2. Note that 3, 5, 7, 13, 31 are the only prime factors of 2^12 - 1 = 4095, but 1/3 has period 2, 1/5 has period 4, 1/7 has period 3, 1/31 has period 5, so 13 is a unique-period prime in base 2. (For the same reason, 13 is a unique-period prime in base 4.)
1/13 has period 3 in base 3. Note that 2, 13 are the only prime factors of 3^3 - 1 = 26, but 1/2 has period 1, so 13 is a unique-period prime in base 3.
1/13 has period 3 in base 22. Note that 3, 7, 13 are the only prime factors of 22^3 - 1 = 10647, but 1/3 and 1/7 both have period 1, so 13 is a unique-period prime in base 22.
PROG
(PARI)
p = 13;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));
CROSSREFS
Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).
Numbers n such that 2^n-1 has only one primitive prime factor, sorted according to the magnitude of the corresponding prime.
+10
2
2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, 26, 42, 13, 34, 40, 32, 54, 17, 38, 27, 19, 33, 46, 56, 90, 78, 62, 31, 80
COMMENTS
Periods associated with A144755 in base 2. The binary analog of A051627.
EXAMPLE
2^12 - 1 = 4095 = 3 * 3 * 5 * 7 * 13, but none of 3, 5, 7 is a primitive prime factor, so the only primitive prime factor of 2^12 - 1 is 13.
MATHEMATICA
nmax = 65536; primesPeriods = Reap[Do[p = Cyclotomic[n, 2]/GCD[n, Cyclotomic[n, 2]]; If[PrimeQ[p], Print[n]; Sow[{p, n}]], {n, 1, nmax}]][[2, 1]]; Sort[primesPeriods][[All, 2]]
Numbers k such that A019320(k) is greater than A064078(k) and the latter is a prime or a prime power.
+10
1
18, 20, 21, 54, 147, 342, 602, 889, 258121
COMMENTS
The unique prime factor of A064078(k) is then a unique prime to base 2 (see A161509), but not a cyclotomic number. In all known examples, A064078(k) is a prime. If A064078(k) was a prime power p^j with j>1, then p would be both a Wieferich prime ( A001220) and a unique prime to base 2. Subsequence of A093106 (the characterization of A093106 can be useful when searching for more terms). Should this sequence be infinite?
PROG
(PARI) for(n=1, +oo, c=polcyclo(n, 2); c % n < 2 && next(); c/=(c%n); ispseudoprime(if(ispower(c, , &b), b, c))&&print1(n, ", "))
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