On a conjecture of Sun

2021/10/10 by Krishnamoorthy, Srilakshmi, Sarma, Abinash
#11D85 #11E25 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/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(a₁,⋯,a_k;n) and T(a₁,⋯,a_k;n) denote the number of ways n can be expressed as ∑_i=1^k a_ix_i² and ∑_i=1^k a_i(x_i(x_i+1))/(2), respectively. Z.-H. Sun, in \cite4, 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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