Problem

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

Tags: quadratics, greatest common divisor, number theory, Integers, diophantine



For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation: \[ x^2 - xy + y^2 = n. \]a) Determine $f(2022)$. b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.