Ternary Sums of Squares and Triangular Numbers

Details Ternary Sums of Squares and Triangular Numbers Ternary Sums of Squares and Triangular Numbers

2010-09-27

arxiv.org/abs/1009.5402v4

Mathematics

Number Theory

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Scientific paper

For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(\alpha, \beta, \gamma)$ for which the ternary sums $\alpha x^2 +\beta T_y + \gamma T_z$ 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.

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