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.