# Pépin’s theorem

Theorem (Pépin). A Fermat number $F_{n}=2^{2^{n}}+1$ is prime only if

 $3^{\frac{F_{n}-1}{2}}\equiv-1\mod F_{n}.$

In other words, if 3 raised to the largest power of two not greater than the Fermat number leaves as a remainder the next higher power of two when divided by that Fermat number (since $F_{n}-1=2^{2^{n}}$), then that Fermat number is a Fermat prime.

For example, $17=2^{2^{2}}+1$ is a Fermat prime, and we can see that $3^{8}=6561$, which leaves a remainder of 16 when divided by 17. The smallest Fermat number not to be a prime is 4294967297, as it is the product of 641 and 6700417, and $3^{2147483648}$ divided by 4294967297 leaves a remainder of 10324303 rather than 4294967296.

Title Pépin’s theorem PepinsTheorem 2013-03-22 18:53:09 2013-03-22 18:53:09 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Theorem msc 11A51 Pepin’s theorem