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Find all pairs yielding a square

Determine all pairs of positive integers $$m,n$$ such that $$2^m + 3^n$$ is a perfect square.

Source: Problem Primer for the Olympiad

Note by Tim Vermeulen
4 years ago

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A brief sketch: Considering modulo $$3, m$$ must be even. Considering modulo $$4, n$$ must be even. Thus we have a primitive pythagorean triple, and so we have $$2xy = 2^{m/2}$$ and $$x^2- y^2 = 3^{n/2}$$. This I believe is easily solved and leads to the only solution being $$m=4,n=2$$ (If I haven't made a mistake). · 4 years ago

But also m=0 and n=1 is a solution !! · 4 years ago

The problem asks for "all pairs of positive integers $$m,n$$". The number $$0$$ is not positive. · 4 years ago

That's right, well done :) · 4 years ago

Thank you for posting such a nice problem! · 4 years ago

The book I got it from has many of the like, I'd definitely recommend it! · 4 years ago