Proving the uncountability of the number of irrational powers of irrational numbers evaluated as rationals and solutions approximation for x^x=y and x^{x^x}=y

TL;DR

This paper proves the uncountability of irrational number pairs where an irrational power yields a rational number and provides estimates for solutions to equations like x^x=y and x^{x^x}=y.

Contribution

It establishes the uncountability of such irrational pairs and offers bounds for solving the equations x^x=y and x^{x^x}=y for given y.

Findings

Uncountably many pairs of irrational numbers produce rational powers.

Constructive proof of irrational power of irrational number resulting in rational.

Bounds for solutions to x^x=y and x^{x^x}=y for real y.

Abstract

Numbers are often used to define more complicated numbers. For example, two integers are used to define a rational number and two reals are used to define a complex number. It might be expected that an irrational power of an irrational number would be an irrational number. Despite that, it is actually possible for an irrational power of an irrational number to be a rational number. For many generations, this circulated as a non-constructive proof by contradiction in logic for discrete mathematics textbooks and college courses. Since the 80s, a constructive proof circulated orally, such as $\sqrt{2}^{log_\sqrt{2} 3}$ equals to 3. A written proof was published in 2008 by Lord. The first contribution of this paper is to prove that there is an uncountable number of such pairs of irrational numbers such that the power of one to the other is a rational number. Marshall and Tan answered the question of whether there is a single irrational number $a$ such that $a^a$ is rational. They proved that given $I$ = $(\frac{1}{e})^{(\frac{1}{e},\infty)}$, then every rational number in $I$ is either of the form $a^a$ for an irrational $a$ or is in the very thin set $\{1,4,27,256,...,n^n,...\}$. It seems a challenging task to analytically solve the equation $x^x=y$ for any real $y$. To the best of our knowledge, there is no work on finding $x$ from a given $y$. We proved that $ln((ln(y))<x$ for $y>e$ and $x<ln(y)$, for $y> e^e$. Hence, the second contribution of this paper is to estimate the real $x$ such that either one of the equations $x^x=y$ or $x^{x^x}=y$ holds, for a given $y$.