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Product is not an integer power

Source: Austrian Federal Competition 2013, part 1, problem 1

June 18, 2013
modular arithmeticnumber theory proposednumber theory

Problem Statement

Show that if for non-negative integers mm, nn, NN, kk the equation (n2+1)2kā‹…(44n3+11n2+10n+2)=Nm(n^2+1)^{2^k}\cdot(44n^3+11n^2+10n+2)=N^m holds, then m=1m = 1.
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