E175 -- Decouverte d'une loi tout extraordinaire des nombres par rapport a la somme de leurs diviseurs

(Discovery of an extraordinary law of numbers in relation to the sum of their divisors)

Summary:

This paper was apparently published three times: Fuss didn't know it had been published in Bibliothèque impartiale when he put it in the Commentationes artihmeticae in 1849 and then in the Opera Postuma in 1862.

Euler again uses S(n) to denote the sum of the divisors of n and presents the various properties of the function. It's possible that much of the material here is also in E152, though this paper also has some properties of partitions (starting in section 9).

According to C. G. J. Jacobi, a treatise with this title was read to the Berlin Academy on June 22, 1747.

Publication:

Originally published in Bibliotheque impartiale 3, 1751, pp. 10-31
Opera Omnia: Series 1, Volume 2, pp. 241 - 253
According to Jacobi, the manuscript of the printed treatise can be found in the archive of the Berlin Academy.
Reprinted in Commentat. arithm. 2, 1849, pp. 639-647 [E175a]
Reprinted in Opera Postuma 1, 1862, pp. 76-84 [E175b]

Documents Available:

Original publication: E175
This paper was translated into English by George Póyla in Mathematics and Plausible Reasoning, vol I, Princeton Univ. Press (1954), pp. 90-96. It was reprinted in the MAA book Biscuits of Number Theory, edited by Art Benjamin and Ezra Brown (2009).
A more recent English translation has been done by Todd Doucet: E175.

Return to the Euler Archive