Question 63
Let $a$, $b$, $c$, $n$ be four integers, where $n \ge 2$, and let $p$ be a prime dividing both $a^2 + ab + b^2$ and $a^n + b^n + c^n$, but not $a + b + c$; for instance, $a \equiv b \equiv -1 \pmod{3}$, $c \equiv 1 \pmod{3}$, $n$ a positive even integer, and $p = 3$ or $a = 4$, $b = 7$, $c = -13$, $n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p-1$ are not coprime.