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equation in 3 variables with 2016, infinitely many solutions

Source: MEMO 2016 T8

August 25, 2016
number theorynumber theory proposedequationmodular arithmetic

Problem Statement

For a positive integer nn, the equation a2+b2+c2+n=abca^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)(a, b, c) for n=2017n = 2017. 2. For n=2016n = 2016, aa is divisible by 33 for all solutions (a,b,c)(a, b, c). 3. There are infinitely many solutions (a,b,c)(a, b, c) for n=2016n = 2016.
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