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
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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

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Manuscript history

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

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Cite this paper

Shiue, P. J.-S., Shannon, A. G., Huang, S. C., & Reyes, J. E. (2022). Notes on efficient computation of Ramanujan cubic equationя. Notes on Number Theory and Discrete Mathematics, 28(2), 350-375, DOI: 10.7546/nntdm.2022.28.2.350-375.

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