A145040 Primes p such that 2^p-1 is prime and congruent to 31 mod 5!.

5, 13, 17, 61, 89, 521, 2281, 3217, 4253, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 132049, 859433, 1398269, 2976221, 3021377, 6972593, 13466917, 30402457, 32582657, 42643801, 43112609, 57885161

Offset: 1

Author

Artur Jasinski, Sep 30 2008

Keywords

• nonn
• more

Comments

Mersenne numbers (with the exception of the first one) are congruent to 7 or 31 mod 5!. This sequence is a subsequence of A000043.

Is this 2 together with the terms of A112634? - R. J. Mathar, Mar 18 2009

Yes. An odd index p > 2 will be congruent to either 1 or 3 mod 4. If it is 1, then 2^p = 2^(4k+1) will be congruent to 2 mod 5, to 0 mod 4, and to 2 mod 3. This completely determines 2^p (and hence 2^p - 1) mod 5!. The other case, when p is congruent to 3 mod 4, will make 2^p congruent to 3 mod 5, to 0 mod 4, and to 2 mod 3. This leads to the other (distinct) value of 2^p mod 5!. This proves that this sequence is just A112634 without the initial term 2. - Jeppe Stig Nielsen, Jan 02 2018

From Jinyuan Wang, Nov 24 2019: (Start)

2^a(n) - 1 is congruent to 1 mod 5 since a(n) is congruent to 1 mod 4, so 5^(2^(a(n)-1) - 1) == (5, 2^a(n) - 1) == (2^a(n) - 1, 5)*(-1)^(2^a(n) - 1) == 1 (mod 2^a(n) - 1), where (m,p) is the Legendre symbol.

Conjecture: For n > 1, the Mersenne number M(n) = 2^n - 1 is in this sequence iff 5^M(n-1) == 1 (mod M(n)). (End)

Links

• Chris K. Caldwell, The largest known primes. - _R. J. Mathar_, Jul 31 2009

Crossrefs

Cf. A000043, A000668, A112634, A124477, A139484, A145038, A112633, A145041, A145042.

Programs

Mathematica

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 5! ] == 31, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a
Select[MersennePrimeExponent[Range[48]], PowerMod[2, #, 120] == 32 &] (* Amiram Eldar, Oct 19 2024 *)

PARI

isok(p) = isprime(p) && isprime(q=2^p-1) && ((q % 120)==31); \\ Michel Marcus, Jan 06 2018

Formula

a(n) = A112634(n+1). - Jeppe Stig Nielsen, Jan 02 2018

Extensions

42643801 inserted by R. J. Mathar, Jul 31 2009

a(28) from Amiram Eldar, Oct 19 2024