Comment on "Which is greater: $e^{\pi}$ or $\pi^{e}$? An unorthodox physical solution to a classic puzzle"

TL;DR

This paper critiques a recent physics-based argument resolving whether $e^{}$ is greater than $^{e}$, emphasizing its limited scope and independence from the second law of thermodynamics.

Contribution

It analyzes and challenges the validity and scope of a physical argument used to compare $e^{}$ and $^{e}$, highlighting its limitations.

Findings

The argument does not depend on the numerical value of .

It is based on the inequality $e^{x} \u2265 x^{e}$ for positive .

The argument is a limited proof of the second law for this specific case.

Abstract

In a recent Note (Am. J. Phys. 92:397, 2024; arXiv:2309.10826), Vallejo and Bove provide a physical argument based nominally on the second law of thermodynamics as a way of resolving the mathematical question appearing in the title. A remarkable aspect of their argument is that it does not depend on the numerical value of $\pi$, because $e^{x} \ge x^{e}$ for all positive $x$, with equality occurring only when $x = e$. Moreover, their argument does not depend on the validity of the second law but is rather a limited proof of it for this particular case.