A demonstration of a theorem on the order observed in the sums of divisors
Euler proves that the infinite product s = (1–x)(1–x2)(1–x3)... expands into the power series s = 1 – x – x2 + x5 + x7 – ..., in which the signs alternate in twos 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. (From Jordan Bell's translation summary.)
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 5, pp. 75-83.
Opera Omnia Citation
Series 1, Volume 2, pp.390-398.