2005/05/18 by Leonhard Euler, Euler, Leonhard
Mathematics · #01A50 #11B37 #FOS: Mathematics #History and Overview (math.HO) #Number Theory (math.NT) #math.HO #math.NT #msc:01A50 #msc:11B37
paper · pdf · doi:10.48550/arxiv.math/0505373
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
arxiv created 2005/05/18 · arxiv updated 2009/12/01
In this paper Euler considers the properties of the pentagonal numbers, those numbers of the form (3n2 ± n)/(2). He recalls that the infinite product (1-x)(1-x2)(1-x3)... 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.