Divisor of Fermat Number/Refinement by Lucas

Theorem
Let $F_n$ be a Fermat number.

Let $m$ be divisor of $F_n$. Let $n \ge 2$.

Then $m$ is in the form:
 * $k \, 2^{n + 2} + 1$

Proof
It is sufficient to prove the result for prime divisors.

The general argument for all divisors follows from the argument:
 * $\paren {a \, 2^c + 1} \paren {b \, 2^c + 1} = a b \, 2^{2 c} + \paren {a + b} \, 2^c + 1 = \paren {a b \, 2^c + a + b} \, 2^c + 1$

So the product of two factors of the form preserves that form.

Let $p$ be a prime divisor of $F_n = 2^{2^n} + 1$.

From Euler's Result:
 * $\exists q \in \Z: p = q \, 2^{n + 1} + 1$

Since $n \ge 2$, $q \, 2^{n + 1}$ is divisible by $2^{2 + 1} = 8$.

Hence:
 * $p \equiv 1 \pmod 8$

By Second Supplement to Law of Quadratic Reciprocity:
 * $\paren {\dfrac 2 p} = 1$

so $2$ is a quadratic residue modulo $p$.

Hence:
 * $\exists x \in \Z: x^2 = 2 \pmod p$

We have shown $2^{2^n} \equiv -1 \pmod p$ and $2^{2^{n + 1} } \equiv 1 \pmod p$.

By Congruence of Powers:
 * $x^{2^{n + 1} } \equiv 2^{2^n} \equiv -1 \pmod p$
 * $x^{2^{n + 2} } \equiv 2^{2^{n + 1}} \equiv 1 \pmod p$

From Integer to Power of Multiple of Order, the order of $x$ modulo $p$ divides $2^{n + 2}$ but not $2^{n + 1}$.

Therefore it must be $2^{n + 2}$.

Hence: