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GCD of the Solutions to a Quadratic Diophantine

Source: Indonesian Stage 1 TST for IMO 2022, Test 4 (Number Theory)

December 25, 2021
quadraticsgreatest common divisornumber theoryIntegersdiophantine

Problem Statement

For each natural number nn, let f(n)f(n) denote the number of ordered integer pairs (x,y)(x,y) satisfying the following equation: x2āˆ’xy+y2=n. x^2 - xy + y^2 = n. a) Determine f(2022)f(2022). b) Determine the largest natural number mm such that mm divides f(n)f(n) for every natural number nn.
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