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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 $\gcd(x,y)=1$). $\endgroup$ Jan 10, 2019 at 15:36

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Here are two unconditional results:

  • if $x$ is a power of $2$ then $(x,y)=(2,6)$

  • $\gcd(x,y)=1$ or $2$.


I thought of this problem in 2019 and posted it on MSE (link), as at that time I did not realise this MO post existed. In Servaes' answer in the link, the two results are shown. The answer by W-t-P is a detailed version of Gerry Myerson's comment above.

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  • $\begingroup$ You can link to individual answers and comments. I edited accordingly. $\endgroup$
    – LSpice
    Feb 19 at 0:42
  • $\begingroup$ fascinating! thanks for the link!!!!! $\endgroup$
    – don bright
    Mar 3 at 0:03

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