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On a conjecture of Sun (2110.04799v1)

Published 10 Oct 2021 in math.NT

Abstract: A number of the form $x(x+1)/2$ where $x$ is an integer is called a triangular number. Suppose, $N(a_1,\cdots,a_k;n)$ and $T(a_1,\cdots,a_k;n)$ denote the number of ways $n$ can be expressed as $\sum_{i=1}k a_ix_i2$ 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 $N(a,b,c;8n+a+b+c)$. In this paper, we prove these conjectures using theta function identities. Moreover, we add some new triplets $(a,b,c)$ satisfying these conjectures.

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