Euler and the pentagonal number 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.