Can $x^x$ be a natural number for non-integer $x$?

Question

Does some real non-integral $x$ exist such that $x^x$ equals a natural number?

Thanks, Tom

Answer 1

The function $x^x$ is continuous, and becomes very large for large $x$. It follows by the Intermediate Value Theorem that every integer $n\ge 1$ is $x^x$ for some real $x$.