Almost universal ternary sums of triangular numbers

Authors:
Wai Kiu Chan and Byeong-Kweon Oh

Journal:
Proc. Amer. Math. Soc. **137** (2009), 3553-3562

MSC (2000):
Primary 11E12, 11E20

Published electronically:
June 25, 2009

MathSciNet review:
2529860

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Abstract | References | Similar Articles | Additional Information

Abstract: For any integer , let denote the triangular number . In this paper we give a complete characterization of all the triples of positive integers for which the ternary sums represent all but finitely many positive integers, which resolves a conjecture of Kane and Sun.

**1.**Leonard Eugene Dickson,*History of the theory of numbers. Vol. II: Diophantine analysis*, Chelsea Publishing Co., New York, 1966. MR**0245500****2.**William Duke and Rainer Schulze-Pillot,*Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids*, Invent. Math.**99**(1990), no. 1, 49–57. MR**1029390**, 10.1007/BF01234411**3.**A. G. Earnest,*Representation of spinor exceptional integers by ternary quadratic forms*, Nagoya Math. J.**93**(1984), 27–38. MR**738916****4.**A. G. Earnest and J. S. Hsia,*Spinor norms of local integral rotations. II*, Pacific J. Math.**61**(1975), no. 1, 71–86. MR**0404142****5.**A. G. Earnest and J. S. Hsia,*Spinor genera under field extensions. II. 2 unramified in the bottom field*, Amer. J. Math.**100**(1978), no. 3, 523–538. MR**0491488****6.**A. G. Earnest, J. S. Hsia, and D. C. Hung,*Primitive representations by spinor genera of ternary quadratic forms*, J. London Math. Soc. (2)**50**(1994), no. 2, 222–230. MR**1291733**, 10.1112/jlms/50.2.222**7.**Song Guo, Hao Pan, and Zhi-Wei Sun,*Mixed sums of squares and triangular numbers. II*, Integers**7**(2007), A56, 5. MR**2373118****8.**J. S. Hsia,*Spinor norms of local integral rotations. I*, Pacific J. Math.**57**(1975), no. 1, 199–206. MR**0374029****9.**B. Kane,*On two conjectures about mixed sums of squares and triangular numbers*, J. Combinatorics and Number Theory**1**(2009), no. 1, 77-90.**10.**B. Kane and Z.W. Sun,*On almost universal mixed sums of squares and triangular numbers*, preprint arXiv:0808.2761.**11.**O. T. O’Meara,*Introduction to quadratic forms*, Die Grundlehren der mathematischen Wissenschaften, Bd. 117, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR**0152507****12.**B.-K. Oh and Z.W. Sun,*Mixed sums of squares and triangular numbers. III*, J. Number Theory**129**(2009), 964-969.**13.**Zhi-Wei Sun,*Mixed sums of squares and triangular numbers*, Acta Arith.**127**(2007), no. 2, 103–113. MR**2289977**, 10.4064/aa127-2-1**14.**Z.W. Sun,*A message to number theory mailing list*, April 27, 2008. http://listserv.nodak.edu/cgi-bin/ wa.exe?A2=ind0804&L=nmbrthry&T=0&P=1670.**15.**André Weil,*Number theory*, Birkhäuser Boston, Inc., Boston, MA, 1984. An approach through history; From Hammurapi to Legendre. MR**734177**

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Additional Information

**Wai Kiu Chan**

Affiliation:
Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459

Email:
wkchan@wesleyan.edu

**Byeong-Kweon Oh**

Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea

Email:
bkoh@math.snu.ac.kr

DOI:
https://doi.org/10.1090/S0002-9939-09-09990-0

Received by editor(s):
August 28, 2008

Published electronically:
June 25, 2009

Additional Notes:
The work of the second author was supported by the Korea Research Foundation Grant (KRF-2008-314-C00004) funded by the Korean Government.

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2009
American Mathematical Society