Problem

Source: INAMO Shortlist 2015 N8

Tags: number theory, Diophantine Equations, infinitely many solutions, Indonesia



The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$. (a) Show that there are infinitely many good numbers. (b) Show that if $n$ is a good number, then $7 \nmid n$.