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A354167
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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.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 16.1.
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LINKS
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CROSSREFS
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Cf. A123271 (sign of the penultimate term of the Lucas-Lehmer sequence).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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