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

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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References

  1. C. Adiga, S. Cooper, and J. H. Han, A general relation between sums of squares and sums of triangular numbers, Int. J. Number Theory, 1 (2005), 175–182.

    MathSciNet  Article  Google Scholar 

  2. A. Alaca, Representations by quaternary quadratic forms whose coefficients are 1, 3 and 9, Acta Arith., 136 (2009), 151–166.

    MathSciNet  Article  Google Scholar 

  3. A. Alaca, Representations by quaternary quadratic forms whose coefficients are 1, 4, 9 and 36, J. Number Theory, 131 (2011), 2192–2218.

    MathSciNet  Article  Google Scholar 

  4. A. Alaca, Ş. Alaca, M. F. Lemire, and K. S. Williams, Nineteen quaternary quadratic forms, Acta Arith., 130 (2007), 277–310.

    MathSciNet  Article  Google Scholar 

  5. A. Alaca, Ş. Alaca, M. F. Lemire, and K. S. Williams, Jacobis identity and representations of integers by certain quaternary quadratic forms, Int. J. Modern Math., 2 (2007), 143–176.

    MathSciNet  MATH  Google Scholar 

  6. A. Alaca, Ş. Alaca, M. F. Lemire, and K. S. Williams, Theta function identities and representations by certain quaternary quadratic forms II, Int. Math. Forum, 3 (2008), 539–579.

    MathSciNet  MATH  Google Scholar 

  7. A. Alaca, Ş. Alaca, M. F. Lemire, and K. S. Williams, Theta function identities and representations by certain quaternary quadratic forms, Int. J. Number Theory, 4 (2008), 219–239.

    MathSciNet  Article  Google Scholar 

  8. A. Alaca, Ş. Alaca, M. F. Lemire, and K. S. Williams, The number of representations of a positive integer by certain quaternary quadratic forms, Int. J. Number Theory, 5 (2009), 13–40.

    MathSciNet  Article  Google Scholar 

  9. N. D. Baruah, S. Cooper, and M. Hirschhorn, Sums of squares and sums of triangular numbers induced by partitions of 8, Int. J. Number Theory, 4 (2008), 525–538.

    MathSciNet  Article  Google Scholar 

  10. N. D. Baruah and M. Kaur, Resolution of some conjectures posed by Zhi-Hong Sun on relations between sums of squares and sums of triangular numbers, submitted.

  11. B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer, New York (1991).

    Book  Google Scholar 

  12. S. Cooper, On the number of representations of integers by certain quadratic forms, Bull. Austral. Math. Soc., 78 (2008), 129–140.

    MathSciNet  Article  Google Scholar 

  13. S. Cooper, On the number of representations of integers by certain quadratic forms, II, J. Combin. Number Theory, 1 (2009), 153–182.

    MathSciNet  MATH  Google Scholar 

  14. M. D. Hirschhorn, The power of q: A personal journey, developments in mathematics, 49, Springer, Cham (2017).

  15. M. Kim and B.-K. Oh, The number of representations by a ternary sum of triangular numbers, J. Korean Math. Soc., 56 (2019), 67–80.

    MathSciNet  MATH  Google Scholar 

  16. Z.-H. Sun, Some relations between t(a; b; c; d; n) and N(a; b; c; d; n), Acta Arith., 175 (2016), 169–189.

    MATH  Google Scholar 

  17. Z.-H. Sun, Ramanujan’s theta functions and sums of triangular numbers, Int. J. Number Theory, 15 (2019), 969–989.

    MathSciNet  Article  Google Scholar 

  18. M. Wang and Z.-H. Sun, On the number of representations of n as a linear combination of four triangular numbers, Int. J. Number Theory, 12 (2016), 1641–1662.

    MathSciNet  Article  Google Scholar 

  19. M. Wang and Z.-H. Sun, On the number of representations of n as a linear combination of four triangular numbers II, Int. J. Number Theory, 13 (2017), 593–617.

    MathSciNet  Article  Google Scholar 

  20. K. S. Williams, n = Δ + Δ + 2(Δ+Δ), Far East J. Math. Sci., 11 (2003), 233–240.

    Google Scholar 

  21. K. S. Williams, On the representations of a positive integer by the forms x2 + y2 + z2 + 2t2 and x2 + 2y2 + 2z2 + 2t2, Int. J. Modern Math., 3 (2008), 225–230.

    MathSciNet  Google Scholar 

  22. K. S. Williams, Number theory in the spirit of Liouville, Cambridge Univ. Press, New York (2011).

    MATH  Google Scholar 

  23. E. X. W. Xia and Z. X. Zhong, 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 (2018), 1–18.

    MathSciNet  Article  Google Scholar 

  24. O. X. M. Yao, 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 (2017), 125–143.

    MathSciNet  Article  Google Scholar 

  25. O. X. M. Yao, 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 (2019), 639–654.

    MathSciNet  Article  Google Scholar 

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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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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