The question of the title is a famous mathematical puzzle and can be addressed by several approaches. 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.

Usually, we use mathematics to solve physics problems. This brief note provides an example of the reverse: A physical law (the second law of thermodynamics) is employed to solve a mathematical puzzle (the title of this paper).

This proof may surprise and interest students, motivating them to explore thermodynamic paths to generate other algebraic inequalities, which results in an improvement in their skills in performing entropy analyses.

The proposed solution is as follows. When two bodies *A* and *B* at different temperatures are placed in contact, energy flows between them until thermal equilibrium are reached. Let us suppose that *A* is an incompressible solid with constant heat capacity *C*, initially at temperature $ T 1 = \pi $, and *B* is an ideal thermal reservoir at temperature $ T 2 = e$, where *e* is Euler's number; both temperatures measured in the same absolute temperature scale.^{1} We will assume that each system exchanges heat only with the other.

*C*> 0, Eq. (4) yields

^{2}This is known as

*Gelfond's constant*, and it is a transcendental number whose approximate value is $ e \pi \u2243 23.14069$. On the other hand, $ \pi e \u2243 22.459 \u2009 15$, and it is unknown whether or not it is transcendental.

^{3}

^{4}after transients have died out, the system will reach a new equilibrium state with the bath at pressure

*P*

_{2}and at the initial temperature. Using the fact that the entropy change of the gas is $ R \u2009 ln \u2009 ( P 1 / P 2 )$, where

*R*is the gas constant, and noting that the heat exchange equals the work done on the gas, we obtain that the entropy change of the universe is

*x*, analogous reasoning allows us to infer the following more general inequality:

Instructors interested in this approach can find thermodynamic derivations of other famous inequalities in Refs. 8–10 (inequalities between means), Ref. 11 (Jensen's inequality), or Ref. 12 (Bernoulli's inequalities and bounds for the logarithmic function).

## ACKNOWLEDGMENTS

This work was partially supported by the Agencia Nacional de Investigación e Innovación and Programa de Desarrollo de las Ciencias Básicas (Uruguay).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## REFERENCES

*Pickover, Wonders of Numbers*