<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] If $n$ is odd and factors nontrivially as $ab$ with $a\le b$, then $n$ is the sum of $a$ consecutive odd numbers of which $b$ is the center one. Since the odd numbers are the first differences of the sequence of perfect squares, this means that $n$ is $(x+a)^2-x^2$ for some integer $x$.
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