Solutions to a quadratic diophantine modular equation

Question

I wonder if solutions are known for this quadratic diophantine modular equation: x²=y² mod (p1 p2) where p1,p2 are given primes and x,y are integers and unknowns?

Answer 1

We assume that the $p_i$ are distinct, and deal with the case the $p_i$ are odd. The argument for the case one of the primes, say $p_1$, is $2$ is essentially the same.

We are told that $p_1p_2$ divides $(x-y)(x+y)$. Perhaps (i) $p_1p_2$ divides $x-y$. That gives the solution $x\equiv y\pmod{p_1p_2}$. Or perhaps (ii) $p_1p_2$ divides $x+y$. That gives the solution $x\equiv -y\pmod{p_1p_2}$.

Or perhaps (iii) $p_1$ divides $x-y$ and $p_2$ divides $x+y$, or perhaps (iv) $p_1$ divides $x+y$ and $p_2$ divides $x-y$.

For case (iii) we have $x\equiv y\pmod{p_1}$ and $x\equiv -y\pmod{p_2}$. So $y=x+tp_1$ for some $t$, and $x+y=2x+tp_1\equiv 0\pmod{p_2}$. Let $x$ be arbitrary in the interval $[0.p_1p_2-1]$, and solve the linear congruence $tp_1\equiv -2x\pmod{p_2}$.