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Prove divisibility by a given expression

Source: Irish Math Olympiad 2014 Q2

March 14, 2017
number theorynumber theory unsolvedDivisibilityinduction

Problem Statement

Prove that for N>1N>1 that (N2)2014āˆ’(N11)106(N^{2})^{2014} - (N^{11})^{106} is divisible by N6+N3+1N^6 + N^3 +1
Is this just a proof by induction or is there a more elegant method? I don't think calculating N=2N = 2 was expected.
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