%0 Generic %1 citeulike:3036290 %A Euler, Leonhard %D 2005 %K Vor1800 available-in-tex-format mathematics number-theory pre1800 %T On the remarkable properties of the pentagonal numbers %U http://arxiv.org/abs/math/0505373 %X In this paper Euler considers the properties of the pentagonal numbers, those numbers of the form \$3n^2 n2\$. 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.

@misc{citeulike:3036290, abstract = {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.}, added-at = {2009-08-02T17:14:35.000+0200}, archiveprefix = {arXiv}, author = {Euler, Leonhard}, biburl = {https://www.bibsonomy.org/bibtex/2cf8a9ebebdd40db54b94cfa7db7e350c/rwst}, citeulike-article-id = {3036290}, citeulike-linkout-0 = {http://arxiv.org/abs/math/0505373}, citeulike-linkout-1 = {http://arxiv.org/pdf/math/0505373}, description = {my bookmarks from citeulike}, eprint = {math/0505373}, interhash = {ffec598eced9609d2c15b872c8a010e6}, intrahash = {cf8a9ebebdd40db54b94cfa7db7e350c}, keywords = {Vor1800 available-in-tex-format mathematics number-theory pre1800}, month = May, posted-at = {2008-07-23 08:50:20}, priority = {2}, timestamp = {2009-08-06T10:36:18.000+0200}, title = {On the remarkable properties of the pentagonal numbers}, url = {http://arxiv.org/abs/math/0505373}, year = 2005 }