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Leyland numbers (named for Paul Leyland) are positive integers of the form $x^y + y^x$ , where x and y are naturals > 1, and also the number 3. The OEIS link is https://oeis.org/A076980

I thought that this might have some nice bunch of Pythagorean triples generated but I was wrong. In the first 5000 such numbers from https://oeis.org/A076980/b076980.txt apparently the only square is $100$.

Are there any other Leyland numbers that are squares? And why not? What sort of mathematics should I study to understand this?

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  • $\begingroup$ Heuristics suggest that $x^y+y^x=z^2$ with $\gcd(x,y)=1$ and $x^{-1}+y^{-1}+2^{-1}<1$ has only finitely many solutions (but it's only a heuristic, and only applies when $\gcd(x,y)=1$). $\endgroup$ – Gerry Myerson Jan 10 at 15:36

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