Problem

Source: INAMO Shortlist 2014 N2

Tags: number theory, composite number, Composite, Diophantine equation



Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$. Show that at least one between $a - 1, b - 1, c -1$ is composite number.