# Is 100 the only Leyland number that is a square?

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?

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

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.