Fermat primes expressed in the form : $x^2+2\cdot y^2$ 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 ?
 A: Based on tests on first $5$ Fermat numbers, I found the following simple identity:
$F_n=2^{2^{n}}+1=(2^{2^{n-1}}-1)^2+ 2 \times (2^{2^{n-2}})^2$.
Thus, take $x=(2^{2^{n-1}}-1)$ and $y=(2^{2^{n-2}})$ and you have $F_n=x^2+2y^2$.
Then I found out this is already known as a recurrence relation satisfied by Fermat numbers.
http://en.wikipedia.org/wiki/Fermat_number
A: The uniqueness follows because $R = \mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain whose only units are $\pm 1.$ As you correctly observe, no rational prime $p$ congruent to $1$ or $3$ (mod 8) remains prime in $R.$ Then there is a prime $z$ in $R$ such that $z \overline{z} = p,$ and the only other primes $w \in R$ with this property are $-z, \overline{z}$ and $-\overline{z}.$ Hence if $z = a + b \sqrt{-2}$ for rational integers $a$ and $b,$ then $p = a^2 + 2b^2$ is the unique expression of $p$ in the form required with $a$ and $b$ positive. The conjecture is true. If $q$ is any prime which divides $F_{n}$ for $n >1,$ then we certainly have $q \equiv 1$ (mod $8$). If $F_n$ is divisible by more than one rational prime, it is a easy matter to combine the corresponding primes from $R$ and their complex conjugates in different ways to produce more than one representation  of $F_n$ in the form $c^2 + 2d^2$ for rational integers $c$ and $d.$ If $F_n$ is a power higher than the first of a single rational prime, a similar argument can be applied.
