Factoring[8 616 460 799] like 100 years ago

[Thomas Puettmann]

Yes, it uses Fermat factorization in various $Z_k$ simultaneously.

The counter counts the number of chain links $n$ that pass the laser beam. A chain with $k$ links is configured in such a way that the laser beam can pass if $8 616 460 699 + n^2$ is a square mod $k$. If the machine stops the human operator must check whether $8 616 460 699 + n^2$ is a square in $Z$.

My machine factors Jevons' number in less than 2 min. But it takes a quarter of an hour to configure the chains...

A very nice paper on a slightly different factoring machine is

J. Shallit, H.C. Williams, F. Morain, Discovery of a Lost Factoring Machine, Mathematical Intelligencer 17 (3), 1995, pp. 41-47.