On the remarkable properties of the pentagonal numbers
| dc.creator | Euler, Leonhard | |
| dc.date | 2005-05-18 | |
| dc.date.accessioned | 2026-07-07T05:19:59Z | |
| dc.date.available | 2026-07-07T05:19:59Z | |
| dc.description | In this paper Euler considers the properties of the pentagonal numbers, those numbers of the form $\frac{3n^2 \pm n}{2}$. He recalls that the infinite product $(1-x)(1-x^2)(1-x^3)...$ expands into an infinite series with exponents the pentagonal numbers, and tries substituting the roots of this infinite product into this infinite series. I am not sure what he is doing in some parts: in particular, he does some complicated calculations about the roots of unity and sums of them, their squares, reciprocals, etc., and also sums some divergent series such as 1-1-1+1+1-1-1+1+..., and I would appreciate any suggestions or corrections about these parts. | |
| dc.description | 16 pages, seems to be first English translation of Euler's Latin original ``De mirabilis proprietatibus numerorum pentagonalium'', Acta Academiae Scientarum Imperialis Petropolitinae 4 (1783), no. 1, 56-75. E542 | |
| dc.identifier | https://arxiv.org/abs/math/0505373 | |
| dc.identifier | http://arxiv.org/abs/math/0505373 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75228 | |
| dc.subject | History and Overview | |
| dc.subject | Number Theory | |
| dc.subject | 01A50; 11B37 | |
| dc.title | On the remarkable properties of the pentagonal numbers | |
| dc.type | text |