On the remarkable properties of the pentagonal numbers

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.

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