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Sum of fourth powers is prime

Source: India TST 2023 Practice Test 2 P3

July 9, 2023
combinatoricsnumber theory

Problem Statement

Let nn be any positive integer, and let S(n)S(n) denote the number of permutations τ\tau of {1,…,n}\{1,\dots,n\} such that k4+(τ(k))4k^4+(\tau(k))^4 is prime for all k=1,…,nk=1,\dots,n. Show that S(n)S(n) is always a square.
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