Fermat's factorization method: Difference between revisions

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'''[[Fermat]]'s factorization method''' is a representation of an [[even and odd numbers|odd]] [[integer]] as the [[difference of two squares]]:

:<math>N = a^2 - b^2.</math>

That difference is [[algebra]]ically factorable as <math>(a+b)(a-b)</math>; if neither factor equals one, it is a proper factorization of ''N''. Put another way, we are looking for ''a'',''b'' such that ''a''² &equiv; ''b''² (mod ''N''), called a [[congruence of squares]].

Each odd number has such a representation. Indeed, if <math>N=cd</math> is a factorization of ''N'', then