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