= Prime divisor of a Fermat number
{title2=$\operatorname{ord}_p(2)=2^{n+1}$}
If a <prime number> $p$ divides the <Fermat number> $F_n$, it is odd and $2^{2^n}\equiv-1\pmod p$. The <multiplicative order> of $2$ divides $2^{n+1}$ but not $2^n$, so it is exactly $2^{n+1}$. <Fermat's little theorem> then gives $2^{n+1}\mid p-1$. For $n\geq1$, every prime divisor is $1$ modulo $4$. A prime need not divide any Fermat number merely because it is $1$ modulo $4$: order $12$ excludes the prime $13$ from the whole family.
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