What are the speediest algorithms for factoring very large integers
I'm faced with factoring many CONSECUTIVE large integers. What are the speediest algorithms available for factoring ALL the integers between (say) 10^N + 1 and 10^(N+1) + 1, for N = 10, 11, 12, 13, 14, 15, ... ? For each such N, how much time T(N) would be required to factor these integers? Would the same algorithm be used for each N ? Can some of these factorizations be accomplished on a Mac at home ?
What you want to do seems unreachable for me (though I may be wrong) using Sage. Note that storing the results is already non trivial: for $N = 10$, even if you count only one bit by integer (of course irrealistic!) it would already require 10Gb if I did not make mistakes.