Dat Equation (Powers of 2)

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