Note that all primes above 3 are either 1 or 5 mod 6.
Quote:
Originally Posted by science_man_88
I know that according to resource on the divisors of Mersenne numbers of prime index that aren't prime are +1/1 mod 8 and of the form 2kp+1 which limits possible k values depending on the exponent mod 8 . I've look at all divisors of the exceptions to 2^371 so far it seems if p mod 6 =5 or 1 then 2 of the factors of 2^p1 seem to be also the same modulo 6 is this verifiable if so could this be used to further reduce the k values needed to be checked ?

Another way to write 5 mod 6 is 1 mod 6. When p is odd, 2^p1 is 1 mod 6.
If N is 1 mod 6, then there must be an even number of factors that are 1 mod 6 (because the factors mod 6 have to multiply to the number mod 6). As far as I can tell, this can't be used to make it easier to find factors.