Quote:
Originally Posted by science_man_88
it has to be less than the number of primes by at least the number of Sophie Germain primes that are 3 mod 4. As already stated if p is a 3 mod 4 Sophie Germain prime 2p+1 divides Mp. if we could answer such questions completely, especially if in simple form GIMPS would not be needed as we could predict what the next exponent would be and therefore just calculate the exponents upto as high as possible.

71324207525210468041 is already congruent to 1 (mod 4), and no cofactors known either, so there still may be a chance that 2^713242075252104680411 is prime. This is just an example, that I am taking into account though.