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Proofs of Some Conjectures of Z. -H. Sun on Relations Between Sums of Squares and Sums of Triangular Numbers

Indian Journal of Pure and Applied Mathematics volume 51, pages 11–38 (2020)Cite this article

Abstract

Let N(a, b, c, d; n) be the number of representations of n as ax2+by2+cz2+dw2 and T(a, b, c, d, n) be the number of representations of n as \(a\frac{{X(X + 1)}}{2} + b\frac{{Y(Y + 1)}}{2} + c\frac{{Z(Z + 1)}}{2} + d\frac{{W(W + 1)}}{2}\) , where a, b, c, d are positive integers, n, X, Y, Z, W are nonnegative integers, and x, y, z, w are integers. Recently, Z.-H. Sun found many relations between N(a, b, c, d, n) and T(a, b, c, d, n) and conjectured 23 more relations. Yao proved five of Sun’s conjectures by using (p, k)-parametrization of theta functions and stated that six more could be proved by using the same method. More recently, Sun himself confirmed two more conjectures by proving a general result whereas Xia and Zhong proved three more conjectures of Sun by employing theta function identities. In this paper, we prove the remaining seven conjectures. Six are proved by employing Ramanujan’s theta function identities and one is proved by elementary techniques.

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Acknowledgement

The authors would like to thank the referee for his/her helpful comments and suggestions. This work of the third author was supported by the National Research Foundation of Korea (NRF-2019R1A6-A3A01096245). This work of the fourth author was supported by the National Research Foundation of Korea (NRF-2017R1A2B4003758) and (NRF-2019R1A2C1086347).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tezpur University, Napaam, Sonitpur, 784 028, Assam, India

    Nayandeep Deka Baruah & Mandeep Kaur

  2. Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, Korea

    Mingyu Kim

  3. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, Korea

    Byeong-Kweon Oh

Authors
  1. Nayandeep Deka Baruah
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  2. Mandeep Kaur
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  3. Mingyu Kim
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  4. Byeong-Kweon Oh
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Correspondence to Nayandeep Deka Baruah.

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Baruah, N.D., Kaur, M., Kim, M. et al. Proofs of Some Conjectures of Z. -H. Sun on Relations Between Sums of Squares and Sums of Triangular Numbers. Indian J Pure Appl Math 51, 11–38 (2020). https://doi.org/10.1007/s13226-020-0382-z

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  • DOI: https://doi.org/10.1007/s13226-020-0382-z

Key words

  • Sum of squares
  • sum of triangular numbers
  • Ramanujan’s theta function
  • representation of quaternary quadratic forms

2010 Mathematics Subject Classification

  • 11D85
  • 11E20
  • 11E25
  • 33E20