Factor base

In computational number theory, a factor base is a small set of prime numbers commonly used as a mathematical tool in algorithms involving extensive sieving for potential factors of a given integer.

A factor base is a relatively small set of distinct prime numbers P, sometimes together with -1. Say we want to factorize an integer n. We generate, in some way, a large number of integer pairs (x, y) for which ${\displaystyle x\neq \pm y}$, ${\displaystyle x^{2}\equiv y^{2}{\pmod {n}}}$, and ${\displaystyle x^{2}{\pmod {n}}{\text{ and }}y^{2}{\pmod {n}}}$ can be completely factorized over the chosen factor base—that is, all their prime factors are in P.

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