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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.

%I A354168 #53 Jul 10 2025 11:04:15
%S A354168 7,17,19,89,107,521,607,1279,2281,3217,4423,9689,11213,21701,44497,
%T A354168 216091,859433,1257787,24036583,30402457,32582657,42643801,57885161,
%U A354168 74207281,82589933
%N 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.
%C A354168 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.
%C A354168 The Lucas-Lehmer theorem says that M_p is a prime iff b_{p-1} == 0 (mod M_p).
%C A354168 Furthermore, if M_p is a prime, then b_{p-2} is congruent to +- 2^((p+1)/2) (mod M_p).
%C A354168 This partitions the Mersenne prime exponents A000043 into two classes, listed here and in A354167.
%D A354168 J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 16.1.
%H A354168 Mersenneforum, <a href="https://mersenneforum.org/showpost.php?p=502204&amp;postcount=39">data for all known Mersenne penultimate residues (up to M#51)</a>
%H A354168 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test#Sign_of_penultimate_term">Lucas-Lehmer primality test. Sign of penultimate term</a>
%Y A354168 Cf. A000043, A000668, A354167.
%Y A354168 Cf. A123271 (sign of the penultimate term of the Lucas-Lehmer sequence).
%K A354168 nonn,more
%O A354168 1,1
%A A354168 _N. J. A. Sloane_, Jun 02 2022, based on Section 16.1 of Cosgrave (2022)
%E A354168 Thanks to _Chai Wah Wu_ for several corrections. - _N. J. A. Sloane_, Jun 02 2022
%E A354168 a(16) from _Chai Wah Wu_, Jun 03 2022
%E A354168 a(17)-a(18) from _Chai Wah Wu_, Jun 04 2022
%E A354168 a(19)-a(25) from _Serge Batalov_, Jun 11 2022