### 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 ?