Problem

Source: Turkey TST 2000 P1

Tags: algebra, polynomial, Vieta, number theory unsolved, number theory



$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$ $(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$