Quote:
Originally Posted by axn
Option 3) A few (million) people decides to take one for the team and systematically guess the next mersenne exponent. [There are about 7 million exponents between 74e6 and 200e6, which can be cut to about half thru very basic TF].

You would kill too many people. Better:
sieve (say) the next one billion integer after M74207281 and submit the survivors one by one.
With this you could reach roughly the same effective sieve depth as for Mersenne numbers,
but you will have not decreasing chance what would you get when you submit Mp for larger and larger p values.
Or maybe even better: sieve on (say) x^1024+1 numbers, these has got prime divisors of p=2 or p=2048*k+1, and you can sieve it more efficiently.