When are Multiples of Polygonal Numbers again Polygonal Numbers?

TL;DR

This paper investigates when multiples of polygonal numbers are again polygonal numbers, revealing differences from triangular numbers, providing conditions for solutions, and showing finiteness results for certain systems of relations.

Contribution

It extends known results about triangular numbers to higher polygonal numbers, providing new conditions and finiteness results for solutions of multiple relations.

Findings

Infinitely many triangular numbers are multiples of other triangular numbers.

Higher polygonal numbers do not always have infinitely many such multiples.

Finitely many solutions exist for systems involving multiple polygonal number relations.

Abstract

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.