On the number of representations of $n=a+b$ with $ab$ a multiple of a polygonal number

TL;DR

This paper investigates how many ways a positive integer can be expressed as a sum of two positive integers, where their product is divisible by a polygonal number, revealing new insights into number representations.

Contribution

It introduces a novel analysis of integer representations constrained by polygonal number divisibility, expanding understanding of additive and multiplicative number theory.

Findings

Derived formulas for the number of such representations

Identified conditions under which representations exist

Provided bounds on the number of representations

Abstract

In this paper, we study the number of representations of a positive integer $n$ by two positive integers whose product is a multiple of a polygonal number.