Rationality and power

Sam Chow, Bin Wei

Published 2014-09-12Version 1

We produce an infinite family of transcendental numbers which, when raised to their own power, become rational. We extend the method, to investigate positive rational solutions to the equation $x^x = \alpha$, where $\alpha$ is a fixed algebraic number. We then explore the consequences of $x^{P(x)}$ being rational, if $x$ is rational and $P(x)$ is a fixed integer polynomial.