Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0505373

On the remarkable properties of the pentagonal numbers

Leonhard Euler
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.HOmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Leonhard Euler. 2005-05-18. On the remarkable properties of the pentagonal numbers. https://arxiv.org/abs/math/0505373

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the Reconstruction of SAS from Other Triangle Congruence Criteria, Part II: Eliminating the Pons Asinorum

In the first part of this work we showed that, within a Hilbert plane deprived of the Side-Angle-Side axiom, the Side-Angle-Angle criterion, together with a ray correspondence principle [\textbf{RCT}], the existence of angle bisectors [\textbf{AB}], the congruence of supplements of congruent angles [\textbf{SA}], and the Pons Asinorum [\textbf{PA}], suffices to reconstruct SAS. We left open the question of whether [\textbf{PA}] is genuinely required alongside the other three principles, noting only a qualitative asymmetry in the nature of the principles involved. In this second part we answer this question: we show that \begin{equation*} \textrm{SAA},\ [\textbf{RCT}],\ [\textbf{AB}] \;\vdash\; [\textbf{PA}], \end{equation*} so that [\textbf{PA}] is redundant among the hypotheses of our main theorem, which improves to \begin{equation*} \textrm{SAA},\ [\textbf{RCT}],\ [\textbf{AB}],\ [\textbf{SA}] \;\vdash\; \textrm{SAS}. \end{equation*} The proof adapts an argument recently given by Donnelly, who reconstructs SAS from SAA together with an angle addition axiom and the existence of angle bisectors.

math.HO↗
arXiv2026

Descartes' Circle Theorem, Princess Elizabeth, and Spinors

In this article, on Descartes' famous "circle theorem," we first attempt to explain how the philosopher could have arrived at the equation that he presents without demonstration in a letter to Princess Palatine Elizabeth of Bohemia in November 1643 (a previously unpublished proof, to our knowledge). Then we report some stages of the subsequent generalization of this theorem, whose Cliffordian flavor, via the equivalence of a quadratic form and the square of a linear form, still mobilizes mathematicians today. This statement, which, over time, has undergone different extensions, Euclidean and non-Euclidean, has found, {\it in fine}, a spinor formalization. We see there the proof of what we may call, in a Bachelardian style, an "inductive value" of the truth, which extends by successive generalizations. We conclude, more briefly, with the contacts between Elizabeth and Descartes, and the perhaps symbolic meaning of these considerations on "kissing circles," as they are called in Anglo-Saxon countries, in the context of their correspondence on the soul and the body, and the question of passions.

math.HO↗
arXiv2026

Mathematical folklore

We present reflections on an experimental seminar studying the human and cultural roots of modern mathematics.

math.HO↗