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 2020-05-17, 07:39 #1 JeppeSN     "Jeppe" Jan 2016 Denmark 2168 Posts Mersenne prime in a Cunningham chain Often when a Mersenne prime M = 2^p - 1 is found, we check also the numbers M - 2 (twin), (M + 2)/3 (Wagstaff) and M*(M + 1)/2 + 1 (right perfect) to make sure M does not have a "companion" prime. In this message, I suggest we check one companion extra, namely 2*M - 1. Because M is prime, it would be possible to do a deterministic test of 2*M - 1, even if the canonical form of 2*M - 1does not reveal it: 2*M - 1 = 2^{p+1} - 3 Because of simple modulo 3 considerations, M cannot be a Sophie Germain prime or a safe prime, because 3 always divides 2*M + 1 and (M - 1)/2 (except for trivial initial examples), so we do not need to check them. And (M + 1)/2 is a power of two. So 2*M - 1 is the only relevant candidate for a Cunningham chain. Here is the status so far: Code:  p: factor of 2^{p+1} - 3 ---------------------------------- 2: prime 3: prime 5: prime 7: 11 13: prime 17: 11 19: prime 31: 9241 61: 37 89: 1823 107: 11 127: 11 521: 193 607: 11 1279: 98807 2203: 47 2281: 131 3217: 11 4253: 23 4423: 313 9689: composite 9941: 419303 11213: composite 19937: 11 21701: composite 23209: 37 44497: 11 86243: 6553 110503: 313 132049: composite 216091: 23 756839: 757 859433: 193 1257787: 11 1398269: 96997 2976221: 1451 3021377: 11 6972593: 103869349 13466917: 11 20996011: 86183 24036583: 59 25964951: 76382783 30402457: 11 32582657: 11 37156667: 11 42643801: 38334482051 43112609: 59 57885161: 193 74207281: unknown! 77232917: 11 82589933: 37463 Of course it would be interesting to resolve the unknown case. More factoring attempts will be needed. A PRP test would take very long. As I said, a deterministic test is possible in principle. It would also be interesting to find factors in the cases where the table above says only "composite". I have submitted all the relevant numbers to factordb.com. /JeppeSN