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

Nayandeep Deka Baruah (), Mandeep Kaur (), Mingyu Kim () and Byeong-Kweon Oh ()
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Nayandeep Deka Baruah: Tezpur University
Mandeep Kaur: Tezpur University
Mingyu Kim: Seoul National University
Byeong-Kweon Oh: Seoul National University

Indian Journal of Pure and Applied Mathematics, 2020, vol. 51, issue 1, 11-38

Abstract: 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.

Keywords: Sum of squares; sum of triangular numbers; Ramanujan’s theta function; representation of quaternary quadratic forms; 11D85; 11E20; 11E25; 33E20 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s13226-020-0382-z

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