MEMO 2016 Problem 8

For a positive integer $n$ , the equation $a^2 + b^2 + c^2 + n = abc$ 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 $n = 2016$ .