On a conjecture of Sun

Srilakshmi Krishnamoorthy; Abinash Sarma

paper

On a conjecture of Sun

arXiv:2110.04799

Abstract

A number of the form $x (x + 1) /2$ where $x$ is an integer is called a triangular number. Suppose, $N (a_{1}, \dots, a_{k}; n)$ and $T (a_{1}, \dots, a_{k}; n)$ denote the number of ways $n$ can be expressed as $\sum_{i = 1}^{k} a_{i} x_{i}^{2}$ and $\sum_{i = 1}^{k} a_{i} \frac{x _{i} ( x _{i} + 1 )}{2}$ , respectively. Z.-H. Sun, in \cite{4}, conjectured some relations between $T (a, b, c; n)$ and $N (a, b, c; 8 n + a + b + c)$ . In this paper, we prove these conjectures using theta function identities. Moreover, we add some new triplets $(a, b, c)$ satisfying these conjectures.