Time for another problem of the week! This one doesn’t necessarily require any number-theoretic knowledge, although a little bit might help point you in the right direction. As usual, let \mathbb{N} be the set of natural numbers, i.e., the set of nonnegative integers.

Describe all functions h: \mathbb{N} \to \mathbb{N} such that h(p^2 + q^2) = h(p)^2 + h(q)^2 for all p, q \in \mathbb{N}.

Please submit solutions to topological[dot]musings[At]gmail[dot]com by Sunday, November 2, 11:59 pm (UTC); do not submit solutions in Comments. Everyone with a correct solution will be inducted into our Hall of Fame! We look forward to your response.