Find all positive integers such that x^y=y^x. Given the graph of x^a and a^x intecepts once at a, it will intercept again and large x as a^x dominates for large x(how could I prove this). Also playing around I have found the solution for x^a=a^a, is a and r^(r/r-1), but this doesn't really answer the question does it as I have defined x=ar, and for a fixed a I can't find x except the obvious solution.
2026-04-02 18:06:16.1775153176
Find all positive integers such that x^y=y^x
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Use $\displaystyle x:=(1+\frac{1}{t})^t$ and $\displaystyle y:=(1+\frac{1}{t})^{t+1}$ which solves the equation $x^y=y^x$ for all $x,y>1$ .
When are $x,y$ positive integers ?