Fermat's factorization method

This is roughly the basis of Fermat's factorization method.

At the same time they should not be too close together, or else the number can be quickly factored by Fermat's factorization method.

Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares:

The methods used to find representations of numbers as sums of two squares are essentially the same as with finding differences of squares in Fermat's factorization method.

Shanks' square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method.

The fundamental ideas of Fermat's factorization method are the basis of the quadratic sieve and general number field sieve, the best-known algorithms for factoring large semiprimes, which are the "worst-case".

The primary improvement that quadratic sieve makes over Fermat's factorization method is that instead of simply finding a square in the sequence of , it finds a subset of elements of this sequence whose product is a square, and it does this in a highly efficient manner.