Problem

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

Tags: number theory



a) Prove that for each positive integer $n$ the number or ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=n$ is finite and is multiple of 6. b) Find all ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=727$.