Ternary universal sums of generalized pentagonal numbers

TL;DR

This paper proves the universality of seven specific ternary sums of generalized pentagonal numbers, confirming Sun's conjectures and completing the classification of such universal sums.

Contribution

The paper establishes the universality of seven previously conjectured ternary sums of generalized pentagonal numbers, filling gaps in Sun's classification.

Findings

Seven quadruples are proven to be universal.

Confirms Sun's conjectures on these quadruples.

Completes the classification of universal ternary sums of generalized pentagonal numbers.

Abstract

For any $m\ge3$, every integer of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}2$ with $x \in \z$ is said to be a generalized $m$-gonal number. Let $a\le b\le c$ be positive integers. For every non negative integer $n$, if there are integers $x,y,z$ such that $n=ap_k(x)+bp_k(y)+cp_k(z)$, then the quadruple $(k;a,b,c)$ is said to be {\it universal}. Sun gave in \cite{s1} all possible quadruple candidates that are universal and proved some quadruples to be universal (see also \cite{gs}). He remains the following quadruples $(5,1,1,k)$ for $k=6,8,9,10$, $(5,1,2,8)$, and $(5,1,3,s)$ for $7 \le s \le 8$ as candidates and conjectured the universality of them. In this article we prove that the remaining 7 quadruples given above are, in fact, universal.