Question 134
We consider the equation $a^{2}+b^{2}+c^{2}+n=a b c$, where $a, b, c$ are positive integers.
Prove:
(a) There are no solutions $(a, b, c)$ for $n=2017$.
(b) For $n=2016$, $a$ must be divisible by $3$ for every solution $(a, b, c)$.
(c) The equation has infinitely many solutions $(a, b, c)$ for $n=2016$.
Prove:
(a) There are no solutions $(a, b, c)$ for $n=2017$.
(b) For $n=2016$, $a$ must be divisible by $3$ for every solution $(a, b, c)$.
(c) The equation has infinitely many solutions $(a, b, c)$ for $n=2016$.