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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). Peiyush Jain · 4 years ago

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@Peiyush Jain But also m=0 and n=1 is a solution !! Mouataz Chadmi · 4 years ago

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@Mouataz Chadmi The problem asks for "all pairs of positive integers \(m,n\)". The number \(0\) is not positive. Jimmy Kariznov · 4 years ago

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@Peiyush Jain That's right, well done :) Tim Vermeulen · 4 years ago

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@Tim Vermeulen Thank you for posting such a nice problem! Peiyush Jain · 4 years ago

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

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

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