A demonstration of a theorem on the order observed in the sums of divisors

Abstract

Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved a recurrence relation for the sum of divisors function. In E244 Euler proves the pentagonal number theorem, and then repeats the proof of the recurrence. In Proposition 1, Euler proves that $\prod_{n=1}^\infty (1-a_n)=1-\sum_{n=1}^\infty a_n(1-a_1)...(1-a_{n-1})$. In Proposition 2 he explicitly states the case $a_n=x^n$. Then in Proposition 3, Euler applies Proposition 2 then collects terms: each time this leaves two terms from the series $\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This is explained in detail in a paper I am working on about Euler's work on the pentagonal number theorem.