E244  Demonstratio theorematis circa ordinem in summis divisorum observatum
(A demonstration of a theorem on the order observed in the sums of divisors)
Summary:
(from a translation by Jordan Bell)
Euler proves that the infinite product s=(1x)(1x^2)(1x^3)... expands into the power
series s=1xx^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.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 7583

Opera Omnia: Series 1, Volume 2, pp. 390  398
 According to Jacobi, the manuscript of the printed treatise can be found in the archive
of the Berlin Academy.
 Reprinted in Commentat. arithm. 1, 1849, pp. 234238 [E244a]
 A handwritten French translation of this treatise can be found in the library of the observatory in
Uccle, near Brussels.
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