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Existence of integer satisfying divisibility condition

Source: Latvian TST for Baltic Way 2019 Problem 14

May 29, 2020
number theory unsolvednumber theory

Problem Statement

Let mm be a positive integer and pp be a prime, such that m2−2m^2 - 2 is divisible by pp. Suppose that there exists positive integer aa such that a2+m−2a^2+m-2 is divisible by pp. Prove that there exists positive integer bb such that b2−m−2b^2- m -2 is divisible by pp.
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