ABSTRACT
The semimechanical method used to make factor tables was the so-called stencil method or movable strip method, which is applicable to many other problems and which developed into the sieve machines of the mid-1920s. If one had no usable factor table, one could use trial divisions to search for small factors of N, the number to be factored. Ideally one would use a list of primes as a source of trial divisors. Sometimes one would use simple arithmetical progressions instead. An alternative procedure was to use a previously prepared product P of small primes and then compute the greatest common divisor of P and N. The success of the Legendre method depends on the fact that if R is a quadratic residue of N, it is also a quadratic residue of every prime factor of N.
