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

Author

Listed:
  • Nayandeep Deka Baruah

    (Tezpur University)

  • Mandeep Kaur

    (Tezpur University)

  • Mingyu Kim

    (Seoul National University)

  • Byeong-Kweon Oh

    (Seoul National University)

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.

Suggested Citation

  • Nayandeep Deka Baruah & Mandeep Kaur & Mingyu Kim & Byeong-Kweon Oh, 2020. "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, Springer, vol. 51(1), pages 11-38, March.
    • Handle: RePEc:spr:indpam:v:51:y:2020:i:1:d:10.1007_s13226-020-0382-z
    DOI: 10.1007/s13226-020-0382-z
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    Cited by:

    1. Nayandeep Deka Baruah & Hirakjyoti Das, 2022. "Relations among representations of integers by certain quadratic forms," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(3), pages 672-682, September.

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