Bounds_of_GCD_for_Sum_and_Difference_Congruent_Squares

Let $x, y, n$ be integers. Let $x \not \equiv \pm y \pmod n$ and $x^2 \equiv y^2 \pmod n$ where $a \equiv b \pmod n$ denotes that $a$ is congruent to $b$ modulo $n$. Then $1 < \gcd \set {x - y, n} < n$ and $1 < \gcd \set {x + y, n} < n$ where $\gcd \set {a, b}$ is the GCD of $a$ and $b$.