Abstract
Recently, Sun posed a number of conjectures on the relations between sums of squares and sums of triangular numbers. Some of these conjectures were confirmed by Baruah, Kaur, Kim, Yao, Xia and Zhong. In this paper, we provide a uniform method to prove all of Sun’s conjectures. Our method is based on a theorem that transforms a conjectural identity of Sun into a generalized eta quotients identity. The later identity is proven automatically by using the MAPLE package thetaids due to Jie Frye and Frank Garvan.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adiga, C., Cooper, S., Han, J.H.: A general relation between sums of squares and sums of triangular numbers. Int. J. Number Theory 1, 175–182 (2005)
Baruah, N.D., Cooper, S., Hirschhorn, M.: Sums of squares and sums of triangular numbers induced by partitions of 8. Int. J. Number Theory 4, 525–538 (2008)
Baruah, N.D., Kaur, M., Kim, M.: Proofs of some conjectures of Z.-H. Sun on relations between sums of squares and sums of triangular numbers. Int. J. Number Theory 51, 11–38 (2020)
Frye, J., Garvan, F.G.: Automatic proof of theta-function identities. In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts Monogr. Symbol. Comput, pp. 195–258. Springer, Cham, (2019)
Hirschhorn, M.D.: The Power of \(q\). Springer, New York (2017)
Kim, Mingyu, Byeong-Kweon, Oh.: The number of representations by a ternary sum of triangular numbers. J. Korean Math. Soc. 56, 67–80 (2019)
Sun, Z.H.: Binary quadratic forms and sums of triangular numbers. Acta Arith. 146, 257–297 (2011)
Sun, Z.H.: Some relations between \(t(a; b; c; d; n)\) and \(N(a; b; c; d; n)\). Acta Arith. 175, 169–189 (2016)
Sun, Z.H.: Ramanujan’s theta functions and sums of triangular numbers. Int. J. Number Theory 15, 969–989 (2019)
Sun, Z.H.: The number of representations of \(n\) as a linear combination of triangular numbers. Int. J. Number Theory 15, 1191–1218 (2019)
Sun, Z.H.: Ramanujan’s theta functions and linear combinations of four triangular numbers. (arXiv:1812.01109v4)
Wang, M., Sun, Z.H.: On the number of representations of \(n\) as a linear combination of four triangular numbers II. Int. J. Number Theory 13, 593–617 (2017)
Xia, E.X.W., Zhang, Y.: Proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers. Int. J. Number Theory 15, 189–212 (2019)
Xia, E.X.W., Zhong, Z.X.: Proofs of some conjectures of Sun on the relations between \(N(a; b; c; d; n)\) and \(t(a; b; c; d; n)\). J. Math. Anal. Appl. 463, 1–18 (2018)
Yao, O.X.M.: The relations between \(N(a, b, c, d;n)\) and \(t(a, b, c, d;n)\) and \((p, k)\)-parametrization of theta functions. J. Math. Anal. Appl. 453, 125–143 (2017)
Yao, O.X.M.: Generalizations of some conjectures of Sun on the relations between \(N(a, b, c, d; n)\) and \(T(a, b, c, d; n)\). Ramanujan J. 48, 639–654 (2019)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11701362 and 11971203), the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20180044) and the Qing Lan Project.
Rights and permissions
About this article
Cite this article
Yao, O.X.M., Liu, E.H. & Bian, M. Automatic proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers. Ramanujan J 59, 365–378 (2022). https://doi.org/10.1007/s11139-022-00568-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-022-00568-2