A354167 Let M_p = 2^p-1 be a Mersenne prime, where p is an odd prime. Sequence lists p such that b_{p-2} == 2^((p+1)/2) mod M_p, where {b_k} is defined in the Comments.

3, 5, 13, 31, 61, 127, 2203, 4253, 9941, 19937, 23209, 86243, 110503, 132049, 756839, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 25964951, 37156667, 43112609, 77232917

Offset: 1

Author

N. J. A. Sloane, Jun 02 2022, based on Section 16.1 of Cosgrave (2022)

Keywords

• nonn
• more

Comments

Let M_p = 2^p-1 (not necessarily a prime) where p is an odd prime, and define b_1 = 4; b_k = b_{k-1}^2 - 2 (mod M_p) for k >= 2.

The Lucas-Lehmer theorem says that M_p is a prime iff b_{p-1} == 0 (mod M_p).

Furthermore, if M_p is a prime, then b_{p-2} is congruent to +- 2^((p+1)/2) (mod M_p).

This partitions the Mersenne prime exponents A000043 into two classes, listed here and in A354168.

References

• J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 16.1.

Links

• Mersenneforum, data for all known Mersenne penultimate residues (up to M#51)
• Wikipedia, Lucas-Lehmer primality test. Sign of penultimate term

Crossrefs

Cf. A000043, A000668, A354168.

Cf. A123271 (sign of the penultimate term of the Lucas-Lehmer sequence).

Extensions

Thanks to Chai Wah Wu for several corrections. - N. J. A. Sloane, Jun 02 2022

a(15) from Chai Wah Wu, Jun 04 2022

a(16)-a(25) from Serge Batalov, Jun 11 2022