---
tag: 1HFRNREDN0
kind: concept
name: Equation xy = yx
aliases: [a-b-b-a, equation-x-y-y-x, equation-x-y, x-y-y-x]
sources:
  - wikipedia: en:50167582@r1329494439
    locator: Equation xy = yx
    note: adopted
    mention: wikipedia://en/50167582
    sections: [Summary]
same_as:
  wikidata: Q23815594
  wikipedia: Equation xy = yx
inventory: [86T9KQ74D2]
---

## Summary

In general, exponentiation fails to be [[commutative|099E72KTF8]]. However, the equation $x^y = y^x$ has an infinity of solutions, consisting of the line $x=y$ and a smooth curve intersecting the line at $(e,e)$, where $e$ is Euler's number. The only integer solution that is on the curve is $2^4=4^2$.