How to prove or disprove following statement :
Conjecture :
Fermat number , $F_n=2^{2^n}+1$ , $(n \geq 2)$ is a prime number iff exists a unique representation of
$F_n$ in the form : $x^2+2\cdot y^2$ , where $\gcd(x,y)=1$ , $x,y \geq 0$ .
Assertion :
For every Fermat number $F_n$ , $(n \geq 2)$ it is true that : $F_n \equiv 1 \pmod 8$ .
Theorem :
Odd prime $p$ is expressible as : $p=x^2+2\cdot y^2$ iff
$p \equiv 1 \pmod 8$ , or $p \equiv 3 \pmod 8$ .
So , it follows that every Fermat prime $F_n$ , $(n \geq 2)$ is expressible as :$F_n=x^2+2\cdot y^2$
Question : How to prove uniqueness of this representation ?