Ternary Sums of Squares and Triangular Numbers

2010/09/27 by Wai Kiu Chan, Anna Haensch, Chan, Wai Kiu +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1009.5402

openalex publication_date 2010/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any integer x, let T_x denote the triangular number (x(x+1))/(2). In this paper we give a complete characterization of all the triples of positive integers (α, β, γ) for which the ternary sums αx² +βT_y + γT_z represent all but finitely many positive integers. This resolves a conjecture of Kane and Sun \cite[Conjecture 1.19(i)]KS08 and complete the characterization of all almost universal ternary mixed sums of squares and triangular numbers.

Citations

• Almost universal ternary sums of triangular numbers

Related

• Almost universal ternary sums of triangular numbers
• Mixed sums of triangular numbers and certain binary quadratic forms
• Universal Sums of Triangular Numbers and Squares
• The number of representations by a ternary sum of triangular numbers
• Almost universal mixed sums of squares and polygonal numbers