Arithmetic Polygons and Sums of Consecutive Squares

TL;DR

This paper introduces arithmetic polygons linked to square pyramidal numbers, proving their existence for all odd N≥3 and infinitely many for even N, expanding understanding of polygonal structures in number theory.

Contribution

It establishes the existence of arithmetic polygons for all odd N≥3 and demonstrates infinitely many for even N, connecting these polygons to square pyramidal numbers.

Findings

Existence of at least one arithmetic polygon with N sides for all odd N≥3

Infinitely many arithmetic polygons with an even number of sides

Connection between arithmetic polygons and triples of square pyramidal numbers

Abstract

We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also show that there are infinitely many arithmetic polygons with an even number of sides.