Possibly the OP might allow Lehmer's "bicycle chain sieves"
<https://en.wikipedia.org/wiki/Lehmer_sieve>,
which for several decades were the state of the art in factoring large numbers;
according to <http://ed-thelen.org/comp-hist/Mike-Williams-Lehmer.html>
the first was made in 1926 and "it was only some time in the early 1960s 
when computers were fast enough to match these speeds".