MathNet 2023-part-03 — 191 problems (EN–VI) Question 27

Let $n$ be any positive integer, and let $S(n)$ denote the number of permutations $\tau$ of $\{1, \dots, n\}$ such that $k^4 + (\tau(k))^4$ is prime for all $k = 1, \dots, n$. Show that $S(n)$ is always a square.