Problem

Source: MEMO 2016 T8

Tags: number theory, number theory proposed, equation, modular arithmetic



For a positive integer n, the equation a2+b2+c2+n=abc is given in the positive integers. Prove that: 1. There does not exist a solution (a,b,c) for n=2017. 2. For n=2016, a is divisible by 3 for all solutions (a,b,c). 3. There are infinitely many solutions (a,b,c) for n=2016.