TITLE: Fermat numbers are coprime
QUESTION [15 upvotes]: So today in my final for number theory I had to prove that the Fermat numbers ($F_n=2^{2^n}+1$) are coprime. 
I know that the standard proof uses the following: $F_n=F_1\cdots F_{n-1}+2$ and then the $\gcd$ divides $2$, and it cannot be two, and hence the numbers are coprime.
However, he asked us to use the hint: "Let $l$ be a prime dividing $F_n$. What can you say about the order of $2$ in $(\mathbb{Z}/l\mathbb{Z})^\times$?" 
I have been thinking about it and I cannot figure out how to use his hint.
Any ideas?

REPLY [4 votes]: Let $A_r=a^{2^r}+1,$
$A_{n+t}=a^{2^{n+t}}+1=(a^{2^n})^{2^t}+1=(A_n-1)^{2^t}+1\equiv2\pmod{A_n}$ for integer $t\ge1$
$\implies(A_{n+t},A_n)=(2,A_n)$
If $a$ is even, $A_r$ will be odd if $r\ge1$
$\implies(2,A_n)=1$ for $n\ge1$