<s>[INST] Odd numbers expressed as : $x^2-y^2$ How to prove following statement :

Conjecture:
An odd number $n$ , $(n>1)$ can be uniquely expressed as : $n= x^2-y^2$ ; $x,y \in \mathbb{Z}^{*}$
if and only if $n$ is a prime number .

If $x-y=m$ , where $m>1$ then $m \mid n$
Proof :
$n=x^2-y^2=(y+m)^2-y^2=y^2+2\cdot y\cdot m +m^2-y^2 \Rightarrow$
$\Rightarrow n=m\cdot (2\cdot y+m) \Rightarrow m \mid n$
Therefore , if $m \neq 1$ it follows that $n$ is a composite number , but how to prove that every odd
composite number ,other than $1$ , has representation : $x^2-y^2$ , where $x-y>1$ ?
 [/INST] Write $n=ab$ with $b>1$ and $x^2-y^2=(x+y)(x-y)$. Solve $x+y=a$, $x-y=b$. This system has integer solutions $x=(a+b)/2$ and $y=(a-b)/2$ because $a$ and $b$ are odd and hence $a\pm b$ is even. Finally, $x-y=b>1$, as required.
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