E244 -- Demonstratio theorematis circa ordinem in summis divisorum observatum

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

(from a translation by Jordan Bell)

Euler proves that the infinite product s=(1-x)(1-x^2)(1-x^3)... expands into the power series s=1-x-x^2+x^5+x^7-..., in which the signs alternate in two's and the exponents are the pentagonal numbers. Euler uses this to prove his pentagonal number theorem, a recurrence relation for the sum of divisors of a positive integer.

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