Proofathon Squares!

Let aa, bb and cc be positive integers that are less than 12151215.Find the number of ordered triples (a,b,c)(a, b, c) such that a2+7b2+5a^2+7b^2+5, b2+7c2+5b^2+7c^2+5 and c2+7a2+5c^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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