2013 Fermat Problem 25

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?

\textbf{(A)}\ 17\quad \textbf{(B)}\ 20\quad \textbf{(C)}\ 19\quad \textbf{(D)}\ 18\quad \textbf{(E)}\ 21

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