# Pepin's test

In 1877 Pepin proved the following theorem for deciding if Fermat numbers are prime (this is one of the nicest examples of the classical primality proving tests):

**Pepin's Test**- Let F
_{n}be a Fermat number. F_{n}is prime if and only if 3^{(Fn-1)/2}≡ −1 (mod F_{n}).

Here 3 can be replaced by any positive integer *k*
for which the Jacobi symbol (*k*|F_{n})
is −1. These include *k* = 3, 5, and 10.

If F_{n} is prime, this primality can
be shown by Pepin's test, but when F_{n} is
composite, Pepin's test does not tell us what the factors
will be (only that it is composite). For example,
Selfridge and Hurwitz showed that F_{14} was
composite in 1963, but it was not until 2010 that its first factor was found.

**See Also:** Fermats, FermatDivisor

**Related pages** (outside of this work)

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