I was looking for pairs $(x,y)$ of positive real numbers which satisfy equations of the form $x^y=y^x, x^{y^x}=y^{x^y}, x^{y^{x^y}}=y^{x^{y^x}},\dots$ etc.
I was able to find solutions for $x^y=y^x$, which can be parametrized as $\left(\left(\frac{n+1}{n}\right)^n,\left(\frac{n+1}{n}\right)^{n+1}\right).$
I used a graphing calculator (Desmos) and it seems that all the equations with odd terms, like $x^{y^x}=y^{x^y}, x^{y^{x^{y^x}}}=y^{x^{y^{x^y}}},\dots$ etc. are only possible for $x=y$. Using methods of calculus, I was able to show that $x<y\implies x^{y^x}<y^{x^y}$, which proves this for the equation with 3 terms. But I can't seem to generalize this further.
I am trying to prove $x<y\implies x^{y^{x^{y^x}}}<y^{x^{y^{x^y}}}$, and to generalize this to all odd term equations of this type.