Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions

TL;DR

This paper investigates which quaternary sums of generalized polygonal numbers are universal, meaning they can represent all nonnegative integers, by applying Ramanujan's theta function identities.

Contribution

It classifies the universality of various quaternary sums involving 3-, 4-, 5-, and 8-gonal numbers using theta function identities.

Findings

Identifies new universal quaternary sums involving generalized polygonal numbers.

Provides a complete classification for sums with r,s,t,u in {3,4,5,8}.

Uses Ramanujan's theta function identities to establish universality.

Abstract

Generalized $m$-gonal numbers are those $p_m(x)= [ (m - 2)x^2 - (m - 4)x ]/2 $ where $x$ and $m$ are integers with $m \geq 3$. If any nonnegative integer can be written in the form $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$, where $a,b,c,d$ are positive integers, then we call $ap_r(h)+bp_s(l)+cp_t(m)+dp_u(n)$ a universal quaternary sum. In this paper, we determine the universality of many quaternary sums when $r,s,t,u \in \{3,4,5,8\}$, using the theory of Ramanujan's theta function identities