Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Some results on $ 3$-cores


Authors: Nayandeep Deka Baruah and Kallol Nath
Journal: Proc. Amer. Math. Soc. 142 (2014), 441-448
MSC (2010): Primary 11P83; Secondary 05A17
DOI: https://doi.org/10.1090/S0002-9939-2013-11784-3
Published electronically: November 4, 2013
MathSciNet review: 3133986
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if $ u(n)$ denotes the number of representations of a nonnegative integer $ n$ in the form $ x^2+3y^2$ with $ x,y\in \mathbb{Z}$, and $ a_3(n)$ is the number of $ 3$-cores of $ n$, then $ u(12n+4)=6a_3(n)$. With the help of a classical result by L. Lorenz in 1871, we also deduce that

$\displaystyle a_3(n)=d_{1,3}(3n+1)-d_{2,3}(3n+1),$

where $ d_{r,3}(n)$ is the number of divisors of $ n$ congruent to $ r$ (mod $ 3$), a result proved earlier by Granville and Ono by using the theory of modular forms and by Hirschhorn and Sellers with the help of elementary generating function manipulations.

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

  • [1] Nayandeep Deka Baruah and Bruce C. Berndt, Partition identities and Ramanujan's modular equations, J. Combin. Theory Ser. A 114 (2007), no. 6, 1024-1045. MR 2337237 (2008e:11126), https://doi.org/10.1016/j.jcta.2006.11.002
  • [2] Nayandeep Deka Baruah, Jonali Bora, and Nipen Saikia, Some new proofs of modular relations for the Göllnitz-Gordon functions, Ramanujan J. 15 (2008), no. 2, 281-301. MR 2377581 (2009a:11036), https://doi.org/10.1007/s11139-007-9079-8
  • [3] Bruce C. Berndt, Ramanujan's notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903 (92j:01069)
  • [4] Bruce C. Berndt, Ramanujan's notebooks. Part V, Springer-Verlag, New York, 1998. MR 1486573 (99f:11024)
  • [5] Wenchang Chu, Common source of numerous theta function identities, Glasg. Math. J. 49 (2007), no. 1, 61-79. MR 2337867 (2008i:33047), https://doi.org/10.1017/S0017089507003424
  • [6] Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and $ t$-cores, Invent. Math. 101 (1990), no. 1, 1-17. MR 1055707 (91h:11106), https://doi.org/10.1007/BF01231493
  • [7] Andrew Granville and Ken Ono, Defect zero $ p$-blocks for finite simple groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 331-347. MR 1321575 (96e:20014), https://doi.org/10.1090/S0002-9947-96-01481-X
  • [8] Michael D. Hirschhorn, Partial fractions and four classical theorems of number theory, Amer. Math. Monthly 107 (2000), no. 3, 260-264. MR 1742127 (2001f:11054), https://doi.org/10.2307/2589321
  • [9] Michael D. Hirschhorn and James A. Sellers, Elementary proofs of various facts about 3-cores, Bull. Aust. Math. Soc. 79 (2009), no. 3, 507-512. MR 2505355 (2010b:11138), https://doi.org/10.1017/S0004972709000136
  • [10] Sarachai Kongsiriwong and Zhi-Guo Liu, Uniform proofs of $ q$-series-product identities, Results Math. 44 (2003), no. 3-4, 312-339. MR 2028683 (2004j:33020)
  • [11] L. Lorenz, Bidrag til tallenes theori, Tidsskrift for Mathematik 3(1) (1871), 97-114.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11P83, 05A17

Retrieve articles in all journals with MSC (2010): 11P83, 05A17


Additional Information

Nayandeep Deka Baruah
Affiliation: Department of Mathematical Sciences, Tezpur University, Sonitpur, PIN-784028, India
Email: nayan@tezu.ernet.in

Kallol Nath
Affiliation: Department of Mathematical Sciences, Tezpur University, Sonitpur, PIN-784028, India
Email: kallol08@tezu.ernet.in

DOI: https://doi.org/10.1090/S0002-9939-2013-11784-3
Keywords: Partitions, $t$-cores, theta functions, dissection
Received by editor(s): March 27, 2012
Published electronically: November 4, 2013
Communicated by: Ken Ono
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society