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^(F_n-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 F₁₄ 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)

• Other "n-1" tests
• Fermat number status page