Weighted sums of generalized polygonal numbers with coefficients 1 or 2

TL;DR

This paper investigates conditions under which weighted sums of generalized polygonal numbers with coefficients 1 or 2 can represent all non-negative integers, extending classical results and analyzing specific cases involving four such numbers.

Contribution

It establishes that for any generalized m-gonal numbers with m ≥ 10, representing 1, m-4, and m-2 suffices to represent all non-negative integers, and studies sums of four such numbers.

Findings

Weighted sums of generalized m-gonal numbers with m ≥ 10 can represent all non-negative integers if they represent 1, m-4, and m-2.

The paper provides conditions for universal representation of integers by these sums.

Analysis of sums of four generalized polygonal numbers with coefficients 1 or 2.

Abstract

In this article, we consider weighted sums of generalized polygonal numbers with coefficients $1$ or $2$. We show that for any $m\ge10$, those weighted sums of generalized $m$-gonal numbers represent every non-negative integers if they only represent $1$, $m-4$, and $m-2$. Furthermore, we study representations of sums of four generalized polygonal numbers with coefficients $1$ or $2$.