Pépin's test

In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named for a French mathematician, Théophile Pépin.

Let ${\displaystyle F_{n}=2^{2^{n}}+1}$ be the nth Fermat number. Pépin's test states that for n > 0,

The expression ${\displaystyle 3^{(F_{n}-1)/2}}$ can be evaluated modulo ${\displaystyle F_{n}}$ by repeated squaring. This makes the test a fast polynomial-time algorithm. However, Fermat numbers grow so rapidly that only a handful of Fermat numbers can be tested in a reasonable amount of time and space.

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