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.
Originally published in Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 75-83
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. 234-238 [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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