Composite Fermat number implies Poulet number

Statement

Suppose $n$ is a nonnegative integer. Let $F_{n}$ be the $n^{th}$ Fermat number:

$F_{n}=2^{2^{n}}+1$.

Then:

$2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}$.

In particular, if $F_{n}$ is composite, then $F_{n}$ is a Poulet number.

Related facts

• Prime divisor of Fermat number is congruent to one modulo large power of two
• Mersenne number for prime or Poulet is prime or Poulet

Proof

Given: A nonnegative integer $n$, $F_{n}=2^{2^{n}}+1$.

To prove: $2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}$.

Proof: For any nonnegative integer $n$, $n+1\leq 2^{n}$. Thus, $2^{n+1}|2^{2^{n}}$.

Now, for a Fermat number $F_{n}$, the order of $2$ modulo $F_{n}$ is precisely $2^{n+1}$. From the above, we conclude that the order of $2$ modulo $F_{n}$ divides $2^{2^{n}}=F_{n}-1$, so $2^{F_{n}-1}\equiv 1{\pmod {F_{n}}}$.