The number of representations by a ternary sum of triangular numbers

TL;DR

This paper investigates the number of solutions to a ternary sum of triangular numbers and establishes relations with representations by quadratic forms, proving several conjectures by Z. H. Sun.

Contribution

It introduces new relations between triangular number representations and quadratic forms, confirming multiple conjectures in the field.

Findings

Established relations between $t(a,b,c;n)$ and quadratic form representations.

Proved several conjectures posed by Z. H. Sun.

Enhanced understanding of the link between triangular numbers and quadratic forms.

Abstract

For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some relations between $t(a,b,c;n)$ and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in \cite{s}.