Rationality and power

TL;DR

This paper constructs an infinite family of transcendental numbers with the property that raising them to their own power yields rational numbers, and explores related algebraic and rational solutions to exponential equations.

Contribution

It introduces a novel method to generate transcendental numbers with specific exponential properties and extends analysis to rational solutions of exponential equations involving algebraic numbers.

Findings

Constructed an infinite family of transcendental numbers with x^x rational

Extended methods to solve x^x=α for algebraic α

Analyzed rational solutions to x^{P(x)} for rational x and polynomial P(x)

Abstract

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.