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Challenge 377: Only two solutions?

Does this equation really only have two solutions?

In the equation below, m and n are both non-negative integers.

(i) Find two possible solutions for m and n. Don't forget that they have to be non-negative integers!

(ii) Either

(a) find a third pair of values for m and n that satisfy this equation.

or (b) explain clearly how you know that there are no more possible values for m and n.