Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On almost universal mixed sums of squares and triangular numbers


Authors: Ben Kane and Zhi-Wei Sun
Journal: Trans. Amer. Math. Soc. 362 (2010), 6425-6455
MSC (2010): Primary 11E25; Secondary 11D85, 11E20, 11E95, 11F27, 11F37, 11P99, 11S99
Published electronically: July 23, 2010
MathSciNet review: 2678981
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than $ 2719$ can be represented by the famous Ramanujan form $ x^2+y^2+10z^2$; equivalently the form $ 2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $ T_z$ denotes the triangular number $ z(z+1)/2$. Given positive integers $ a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ ax^2+by^2+cT_z$ represents sufficiently large integers and to establish similar results for the forms $ ax^2+bT_y+cT_z$ and $ aT_x+bT_y+cT_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $ 2ax^2+y^2+z^2$ if and only if all prime divisors of $ a$ are congruent to 1 modulo 4. (ii) The form $ ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of $ a$ is congruent to 1 or 3 modulo 8. (iii) $ ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4. (iv) When $ v_2(a)\not=3$, the form $ aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4 and $ v_2(a)\not=5,7,\ldots$, where $ v_2(a)$ is the $ 2$-adic order of $ a$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11E25, 11D85, 11E20, 11E95, 11F27, 11F37, 11P99, 11S99

Retrieve articles in all journals with MSC (2010): 11E25, 11D85, 11E20, 11E95, 11F27, 11F37, 11P99, 11S99


Additional Information

Ben Kane
Affiliation: Department of Mathematics, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: bkane@math.uni-koeln.de

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@nju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05290-0
PII: S 0002-9947(2010)05290-0
Keywords: Representations of integers, triangular numbers, sums of squares, quadratic forms, half-integral weight modular forms
Received by editor(s): September 18, 2008
Published electronically: July 23, 2010
Additional Notes: This research was conducted when the first author was a postdoctor at Radboud Universiteit, Nijmegen, Netherlands.
The second author was the corresponding author and he was supported by the National Natural Science Foundation (grant 10871087) of the People’s Republic of China
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.