Fermat's factorization method

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

${\displaystyle N=a^{2}-b^{2}.}$

That difference is algebraically factorable as ${\displaystyle (a+b)(a-b)}$; if neither factor equals one, it is a proper factorization of N.

Each odd number has such a representation. Indeed, if ${\displaystyle N=cd}$ is a factorization of N, then

${\displaystyle N=\left({\frac {c+d}{2}}\right)^{2}-\left({\frac {c-d}{2}}\right)^{2}}$

Since N is odd, then c and d are also odd, so those halves are integers. (A multiple of four is also a difference of squares: let c and d be even.)

In its simplest form, Fermat's method might be even slower than trial division (worst case). Nonetheless, the combination of trial division and Fermat's is more effective than either by itself.