Abstract
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 (α, β, γ) 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 (Trans Am Math Soc 362:6425–6455, 2010, Conjecture 1.19(i)) and complete the characterization of all almost universal ternary mixed sums of squares and triangular numbers.
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Acknowledgements
The first author extends his thanks to the Department of Mathematics in University of Florida for their hospitality during his visit for the Focused Week on Integral Lattices in February 2010. Both authors thank Maria Ines Icaza for her helpful comments and discussion. They also thank the referee for his/her patience and helpful comments.
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Chan, W.K., Haensch, A. (2013). Almost Universal Ternary Sums of Squares and Triangular Numbers. In: Alladi, K., Bhargava, M., Savitt, D., Tiep, P. (eds) Quadratic and Higher Degree Forms. Developments in Mathematics, vol 31. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7488-3_3
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DOI: https://doi.org/10.1007/978-1-4614-7488-3_3
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