Expansion of the infinite product (1-x)(1-xx)(1-x³)(1-x⁴)(1-x⁵)(1-x⁶) etc. into a simple series

2004/11/20 by Leonhard Euler, Euler, Leonhard, Jordan Bell +1
Mathematics · #01A50 #11-03 #33D15 #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.HO #math.NT #msc:01A50 #msc:11-03 #msc:33D15

paper · pdf · doi:10.48550/arxiv.math/0411454

7 pages; E541

openalex publication_date 2004/11/20 · arxiv created 2009/08/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Translated from the Latin original "Evolutio producti infiniti (1-x)(1-xx)(1-x³)(1x⁴)(1-x⁵)(1-x⁶) etc. in seriem simplicem" (1775). E541 in the Enestroem index. In this paper Euler is revisiting his proof of the pentagonal number theorem. He gives his original proof explained a bit differently, and then gives a different proof. However this second proof is still rather close to his original proof. To understand the two proofs, I wrote them out using subscript notation and sum/product notation. It would be a useful exercise to try to really understand the proofs without using any modern notation. The right notation takes care of a lot for us, which we would otherwise have to keep active in our minds.

Related

• Evolutio producti infiniti (1-x)(1-xx)(1-x³)(1-x⁴)(1-x⁵) etc. in seriem simplicem
• On the remarkable properties of the pentagonal numbers
• An observation on the sums of divisors
• A demonstration of a theorem on the order observed in the sums of divisors
• Euler and the pentagonal number theorem