Question 87
a) Show that $m^2 - m + 1$ is an element of the set $\{n^2 + n + 1 \mid n \in \mathbb{N}\}$, for any positive integer $m$.
b) Let $p$ be a perfect square, $p \gt 1$. Prove that there exist positive integers $r$ and $q$ such that $p^2 + p + 1 = (r^2 + r + 1)(q^2 + q + 1)$.
b) Let $p$ be a perfect square, $p \gt 1$. Prove that there exist positive integers $r$ and $q$ such that $p^2 + p + 1 = (r^2 + r + 1)(q^2 + q + 1)$.