A354168 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.

7, 17, 19, 89, 107, 521, 607, 1279, 2281, 3217, 4423, 9689, 11213, 21701, 44497, 216091, 859433, 1257787, 24036583, 30402457, 32582657, 42643801, 57885161, 74207281, 82589933

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 A354167.

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, A354167.

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(16) from Chai Wah Wu, Jun 03 2022

a(17)-a(18) from Chai Wah Wu, Jun 04 2022

a(19)-a(25) from Serge Batalov, Jun 11 2022