Prove that there are infinitely many positive integers $a, b$ and $c$ such that their greatest common divisor is $1$ (ie: $gcd(a, b, c) = 1$) and satisfy that: $$a^2=b^2+c^2+bc$$
Source: 2021 Peru PAGMO TST P5
Tags: number theory
Prove that there are infinitely many positive integers $a, b$ and $c$ such that their greatest common divisor is $1$ (ie: $gcd(a, b, c) = 1$) and satisfy that: $$a^2=b^2+c^2+bc$$