FERMAT 2013 Problem 25

For how many odd integers k between 0 and 100 does the equation

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

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

Answer Choices
A. 17
B. 20
C. 19
D. 18
E. 21