2100.21 – Symmetric Exponents

Find two whole numbers $x$ and $y$ such that $x \neq y$ and $x^y=y^x$ .

Solution

With a bit of guessing and trying, you can find that $2^4=4^2=16$ . Note that while there exist infinitely many solutions to the equation $x^y=y^x$ , (graph the equation to see), this is the only solution where $x$ and $y$ are both whole numbers.