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

TL;DR

This paper offers a novel solution to the classic puzzle of whether e^π is greater than π^e, using the second law of thermodynamics, and extends the approach to more general exponential comparisons.

Contribution

It introduces a thermodynamics-based method to solve a well-known mathematical puzzle and generalizes the approach to broader exponential inequalities.

Findings

e^π is greater than π^e confirmed by thermodynamic reasoning

The method can be extended to compare other exponential functions

Provides an alternative perspective to classical mathematical solutions

Abstract

The question of the title is a famous puzzle in the field of recreational mathematics, and can be addressed by several approaches. A compilation of solutions, some of them very ingenious, can be found in [1]. In this contribution we present an alternative solution based on the second law of thermodynamics. The method can be extended to derive a more general result involving the exponential function.