[Paper Review] Euler and the pentagonal number theorem

Jordan Bell|ArXiv.org|Oct 3, 2005

History and Theory of Mathematics5 references3 citations

TL;DR

This paper provides a comprehensive historical and mathematical analysis of Leonhard Euler's work on the pentagonal number theorem, detailing its derivation, applications to the divisor and partition functions, and connections to divergent series and $q$-series. Euler's original insights are contextualized through his correspondence and publications, with the theorem shown to be a foundational result in $q$-series and theta functions, particularly as a special case of the Jacobi triple product identity.

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.

Motivation & Objective

• To provide a complete historical review of Euler’s correspondence and publications concerning the pentagonal number theorem.
• To examine Euler’s applications of the theorem to the divisor function and partition function using generating functions.
• To explore Euler’s treatment of divergent series related to pentagonal numbers and their summation via finite differences.
• To establish the pentagonal number theorem as a special case of the Jacobi triple product identity and its significance in $q$-series theory.
• To contextualize Euler’s unpublished work on partitions and his broader contributions to analytic number theory and special functions.

Proposed method

• Analyzing Euler’s original letters and publications, including correspondence with Bernoulli, Goldbach, and others, to trace the development of the pentagonal number theorem.
• Deriving the pentagonal number theorem as a formal identity: $\prod_{m=1}^{\infty}(1 - x^m) = \sum_{n=-\infty}^{\infty} (-1)^n x^{n(3n-1)/2}$, using algebraic manipulation and infinite product expansions.
• Applying the Jacobi triple product identity to derive the pentagonal number theorem, showing that $\phi(x) = \prod_{m=1}^{\infty}(1 - x^m)$ converges absolutely for $|x| < 1$.
• Using finite difference tables to assign summability to divergent series such as $\sum (-1)^n p_n$ where $p_n$ are pentagonal numbers, yielding results like $s + t = 0$.
• Interpreting Euler’s method of separating series into components based on roots of unity, leading to identities like $\sum \pm A^\lambda \pm B^\lambda \pm \cdots = 0$ for any integer $\lambda$.
• Connecting the pentagonal number theorem to theta functions and modular forms, particularly via $\vartheta_4(q/6; q) = (q; q)_\infty$ for $q^2/3 = x$.

Experimental results

Research questions

• RQ1 How did Euler originally derive and justify the pentagonal number theorem through his correspondence and publications?
• RQ2 What role did the pentagonal number theorem play in Euler’s work on the partition function and divisor function?
• RQ3 How did Euler assign values to divergent series involving pentagonal numbers using finite differences and roots of unity?
• RQ4 In what way is the pentagonal number theorem a special case of the Jacobi triple product identity?
• RQ5 What was the significance of Euler’s unpublished work on partitions, and how did it anticipate later developments in $q$-series theory?

Key findings

• Euler first encountered the pentagonal number theorem through correspondence with Daniel I Bernoulli in 1740–1741, who noted the series expansion of $\prod (1 - n^{-m})$ but could not prove the pattern.
• Euler’s 1751 paper E158 introduced the generating function $\prod_{x=1}^{\infty} \frac{1}{1 - n^x} = \sum n^{(\infty)} n^x$, linking the pentagonal number theorem to the unrestricted partition function.
• The pentagonal number theorem is formally expressed as $\prod_{m=1}^{\infty}(1 - x^m) = \sum_{n=-\infty}^{\infty} (-1)^n x^{n(3n-1)/2}$, with exponents being generalized pentagonal numbers.
• Euler used finite difference tables to assign values to divergent series: for one series, $s = \frac{3}{16}$, for another $t = -\frac{3}{16}$, so $s + t = 0$, demonstrating a consistent summation method.
• Euler generalized the identity to $\sum \pm A^\lambda \pm B^\lambda \pm \cdots = 0$ for any integer $\lambda$, where $A, B, \dots$ are pentagonal numbers congruent modulo $n$.
• The pentagonal number theorem is a special case of the Jacobi triple product identity, and its cube yields $\prod_{m=1}^{\infty}(1 - x^m)^3 = \sum_{n=0}^{\infty} (-1)^n (2n+1) x^{n(n+1)/2}$.