Dat Equation (Powers of 2)

Level pending

For how many odd integers \(k\) between 0 and 100 does the equation:

\[2^{4m^2}+2^{m^2-n^2+4}=2^{k+4}+2^{3m^2+n^2+k}\]

have exactly two pairs of positive integers \((m, n)\) that are solutions?

Source: CEMC Fermat Contest 2013, Problem 25

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