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Primes squares odd condition

Source: Nigeria Olympiad senior mathematics round 2 problem 4 2020

January 19, 2020
number theory

Problem Statement

Let N>=2N>= 2 be an integer. Show that 4n(Nāˆ’n)+14n(N-n)+1 is never a perfect square for each natural number nn less than NN if and only if N2+1N^2+1 is prime.
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