Proofathon Squares!

Let \(a\), \(b\) and \(c\) be positive integers that are less than \(1215\).Find the number of ordered triples \((a, b, c)\) such that \(a^2+7b^2+5\), \(b^2+7c^2+5\) and \(c^2+7a^2+5\) are all perfect squares.


This problem is from the Proofathon Number Theory Contest.


This problem is from the set "Olympiads and Contests Around the World -3". You can see the rest of the problems here.

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