[jerf]

"Lehmer sieves were very fast, in one particular case factoring 2^93 + 1 in 3 seconds."

Impressive. I wondered how fast my computer could do it. Here's the GNU coreutils factor:

    $ time factor 9903520314283042199192993793
    9903520314283042199192993793: 3 3 529510939 715827883 2903110321

    real    0m0.007s
    user    0m0.004s
    sys     0m0.000s

Still impressive. This led me down factor's documentation to the Pollard Rho algorithm: http://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm