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

1,1

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

Table of n, a(n) for n=1..25.

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).

Sequence in context: A144471 A190667 A062304 * A350392 A281874 A135532

Adjacent sequences: A354164 A354165 A354166 * A354168 A354169 A354170

Keyword

nonn,more

Author

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

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

Status

approved