Euler and the pentagonal number theorem

TL;DR

This paper provides a comprehensive historical review of Euler's work on the pentagonal number theorem, including his correspondence, publications, and applications to various mathematical functions, with translations of key Latin, French, and German texts.

Contribution

It offers the first exhaustive compilation and translation of Euler's work on the pentagonal number theorem and its applications, enriching the historical understanding of this mathematical development.

Findings

Detailed analysis of Euler's correspondence and publications

Translations of Euler's Latin, French, and German texts

Comprehensive overview of Euler's applications of the theorem

Abstract

``In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem to the divisor function, partition function and divergent series. We have attempted to give an exhaustive review of all of Euler's correspondence and publications about the pentagonal number theorem and his applications of it.'' This paper gives an exhaustive summary of Euler's work on the pentagonal number theorem. I have gone through all of Euler's published correspondence (except with du Maupertuis and Frederic II) and his papers to find each time he discusses the pentagonal number theorem or applications of it. I have translated from the Latin many sections of his correspondence and papers that are not available in the English, and also sections in the French and German with Dr. Paul Mezo and Christian Leger.