  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.
