Notes on efficient computation of Ramanujan cubic equations

Peter J.-S. Shiue, Anthony G. Shannon, Shen C. Huang, Jorge E. Reyes
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 2, Pages 350–375
DOI: 10.7546/nntdm.2022.28.2.350-375
Full paper (PDF, 335 Kb)

Details

Authors and affiliations

Peter J.-S. Shiue
Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, NV, 89154, United States of America

Anthony G. Shannon
Warrane College, University of New South Wales
Kensington, NSW 2033, Australia

Shen C. Huang
Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, NV, 89154, United States of America

Jorge E. Reyes
Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, NV, 89154, United States of America

Abstract

This paper considers properties of a theorem of Ramanujan to develop properties and algorithms related to cubic equations. The Ramanujan cubics are related to the Cardano cubics and Padovan recurrence relations. These generate cubic identities related to heptagonal triangles and third order recurrence relations, as well as an algorithm for finding the real root of the relevant Ramanujan cubic equation. The algorithm is applied to, and analyzed for, some of the earlier examples in the paper.

Keywords

• Ramanujan-type identity
• Cubic equation
• Trigonometric functions

2020 Mathematics Subject Classification

• Primary 11C08
• Secondary 11B83

References

1. Bankoff, L., & Garfunkel, J. (1973). The heptagonal triangle. Mathematics Magazine, 46(1), 7–19.
2. Berndt, B. C., & Bhargava, S. (1993). Ramanujan–for lowbrows. The American
   Mathematical Monthly, 100(7), 644–656.
3. Berndt, B. C., & Zaharescu, A. (2004). Finite trigonometric sums and class numbers. Mathematische Annalen, 330(3), 551–575.
4. Berndt, B. C., & Zhang, L.-C. (1992). Ramanujan’s identities for eta-functions.
   Mathematische Annalen, 292(1), 561–573.
5. Chen, W. Y. C. Cubic Equations Through the Looking Glass of Sylvester. College
   Mathematics Journal (to appear).
6. Deveci, Ö., & Shannon, A. G. (2017). Pell–Padovan-circulant sequences and their applications. Notes on Number Theory and Discrete Mathematics, 23(3), 100–114.
7. Dresden, G., Panthi, P., Shrestha, A., & Zhang, J. (2019). Cubic polynomials, linear shifts, and Ramanujan simple cubics. Mathematics Magazine, 92(5), 374–381.
8. Gilbert, L., & Gilbert, J. (2014). Elements of Modern Algebra. Cengage Learning.
9. Hildebrand, F. B. (1956). Introduction to Numerical Analysis. McGraw-HilI Book Co, New York.
10. Leyendekkers, J. V., & Shannon, A. G. (1999). The Cardano family of equations. Notes on Number Theory and Discrete Mathematics, 5(4), 151–162.
11. Liao, H.-C., Saul, M., & Shiue, P. J.-S. (2022). Revisiting the General Cubic: A
    Simplification of Cardano’s Solution. arXiv. https://doi.org/10.48550/arXiv.2204.07507
12. Liu, Z.-G. (2003). Some Eisenstein series identities related to modular equations of the seventh order. Pacific Journal of Mathematics, 209(1), 103–130.
13. Ramanujan, S. (1957). Notebooks of Srinivasa Ramanujan (2 Volumes), Tata Institute of Fundamental Research, Bombay.
14. Shannon, A. G. (1972). Iterative formulas associated with generalized third order recurrence relations. SIAM Journal on Applied Mathematics, 23(3), 364–368.
15. Shannon, A. G., Horadam, A. F., & Anderson, P. G. (2006). The auxiliary equation associated with the plastic number. Notes on Number Theory and Discrete Mathematics, 12(1), 1–12.
16. Shevelev, V. (2009). On Ramanujan cubic polynomials. South East Asian Mathematics and Mathematical Sciences, 2009, 113–122.
17. Sylvester, J. J. (1851). Lx. On a remarkable discovery in the theory of canonical forms and of hyperdeterminants. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(12), 391–410.
18. Vinberg, E. B. (2003). A Course in Algebra, In: Graduate Studies in Mathematics Series, Vol. 56, American Mathematical Society.
19. Wang, K., (2019). Heptagonal Triangle and Trigonometric Identities. Forum
    Geometricorum, 19, 29–38.
20. Wang, K. (2019). Integer Sequences for the Sum of Powers of Trigonometric Values. Open Access Library Journal, 6(5), 1–41.
21. Wang, K. (2019). On cubic equations with zero sums of cubic roots of roots. Preprint. Available online: https://www.researchgate.net/publication/335392159
22. Wang, K. (2021). On Ramanujan type identities and Cardano formula. Notes on Number Theory and Discrete Mathematics, 27(3), 155–174.
23. Wituła, R. (2009). Ramanujan-type trigonometric formulas: The general form for the argument 2π/7. Journal of Integer Sequences, 12, Article 09.8.5.
24. Wituła, R. (2010). Full description of Ramanujan cubic polynomials. Journal of Integer Sequences, 13, Article 10.5.7.
25. Wituła, R. (2012). Ramanujan type trigonometric formulae. Demonstratio Mathematica, 45(4), 779–796.
26. Yiu, P. (2009). Heptagonal triangles and their companions. Forum Geometricorum, 9, 125–148.

Manuscript history

• Received: 27 April 2022
• Revised: 7 May 2022
• Accepted: 7 June 2022
• Online First: 14 June 2022