How to write an integer into a product

Question

of two nontrivial rational numbers ?

Given an integer $n$, suppose that it has two decompositions : $p^2 +q^2 = s^2 +t^2 \ $ ($p,q,s,t$ are all positive and $(s,t) \not \in \{(p,q),(q,p)\} \ $).

We want to write $n$ as a product of two nontrivial rational numbers.

Let us write $p=s+x$ and $q=t-y$ where $x \in \mathbb{Z}$ such that $x\ge -s$ and $y \in \mathbb{Z}$ such that $y \le t$.

Rewriting the equalities we get : $n= s^2 +x^2 +2sx + t^2 +y^2 -2ty$.

Then $0= x^2+y^2 +2sx -2ty$.

In the same way we find that : $n= p^2+x^2- 2px + q^2 +y^2 +2qy$.

Then $0 = x^2 +y^2 -2px +2qy$.

Hence $2sx - 2ty = -2px +2qy \Leftrightarrow sx-ty=-px+qy\Leftrightarrow x(s+p)=y(t+q) \Leftrightarrow x= y\dfrac{t+q}{s+p}$.

If we substitute in $n$,we obtain : $n= p^2 +y^2\left(\dfrac{t+q}{s+p}\right)^2 - 2py\dfrac{t+q}{s+p}+q^2+y^2 + 2qy$.

But I'm stuck, it seems that I'm turning around...

Thanks in advance !