Sums of generalized polygonal numbers of almost prime "length"

TL;DR

This paper investigates the representation of integers as sums of three generalized m-gonal numbers with parameters having a limited number of prime factors, establishing density results under certain modular restrictions.

Contribution

It introduces new density results for sums of generalized polygonal numbers with parameters constrained by prime factor counts, extending classical polygonal number theory.

Findings

Density one set of integers are representable under prime factor restrictions

Representation depends on modular conditions on m

Large squarefree parts of f_m(n) ensure representability

Abstract

In this paper, we consider sums of three generalized $m$-gonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restrictions on $m$ modulo $30$, we show that a density one set of integers is represented as such a sum, where the parameters are restricted to have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$ is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is a natural linear function in $n$.