 Prime divisor of a Fermat number

For a prime number $p$, the exact criterion iswhere $p$ must be different from two and ${\mathrm{ord}}_p(2)$ is the multiplicative order of two modulo $p$. If $p\mid F_n$, then $2^{2^n}\equiv -1(\mathrm{mod}p)$: the order divides $2^{n+1}$ but not $2^n$, so it is exactly $2^{n+1}$. Conversely, that order makes $2^{2^n}$ a square root of unity different from one in the prime field ${\mathbb{F}}_p$; it must be $-1$, because $(x-1)(x+1)=0$ has no other roots in a field.

For example, modulo seventeen the powers of two are $2,4,8,16,15,13,9,1$. Their first return to one occurs after eight steps. The half-way value is $16=-1(\mathrm{mod}17)$, proving that $17\mid F_2$. Likewise, ${\mathrm{ord}}_{641}(2)=64$ identifies $641$ as a prime factor of $F_5$. These are statements about divisibility, not evidence that the entire Fermat number is prime.

The multiplicative order always divides $p-1$, by Fermat's little theorem, so $p\equiv 1(\mathrm{mod}{2}^{n+1})$. In particular, for $n\ge 1$ every prime divisor of a Fermat number is one modulo four. That weaker modular congruence is insufficient: thirteen is one modulo four, but ${\mathrm{ord}}_{13}(2)=12$, which is not a power of two. It divides no Fermat number at any index. The exact order criterion explains the whole family, instead of testing finitely many terms.