mathschallenge.net

Problem

Consider the equation, $x^2 + y^2 = 2z^2$; if $GCD(x,y) = 1$ then the solution is primitive.

For example, $7^2 + 17^2 = 2 \times 13^2$ is a primitive solution, whereas $2^2 + 14^2 = 2 \times 10^2$ is not a primitive solution.

Prove that inifinitely many primitive solutions exist.