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

Source: *Problem Primer for the Olympiad*

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

Source: *Problem Primer for the Olympiad*

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TopNewestA 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). – Peiyush Jain · 4 years ago

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– Mouataz Chadmi · 4 years ago

But also m=0 and n=1 is a solution !!Log in to reply

– Jimmy Kariznov · 4 years ago

The problem asks for "all pairs of positive integers \(m,n\)". The number \(0\) is not positive.Log in to reply

– Tim Vermeulen · 4 years ago

That's right, well done :)Log in to reply

– Peiyush Jain · 4 years ago

Thank you for posting such a nice problem!Log in to reply

– Tim Vermeulen · 4 years ago

The book I got it from has many of the like, I'd definitely recommend it!Log in to reply

This problem is from Indian National Mathematical Olympiad – Arghya Datta · 4 years ago

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