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".