The structure of ‘Pi’

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 4, Pages 61—68
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Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney
Sydney, NSW 2006, Australia

A. G. Shannon
Faculty of Engineering & IT, University of Technology
Sydney, NSW 2007, Australia

Abstract

Classes of the modular ring Z4 were substituted into convergent infinite series for π and √2 to obtain Q, the ratio of the arc of a circle to the side of an inscribed square to yield π = 2√2 Q. The corresponding convergents of the continued fractions for π, √2 and Q were then considered, together with the class patterns of the modular rings {Z4, Z5, Z6} and decimal patterns for π.

Keywords

  • Integer structure analysis
  • Modular rings
  • Prime numbers
  • Fibonacci numbers
  • Arctangents
  • Infinite series
  • Pell sequence
  • Continued fractions
  • Triangular numbers

AMS Classification

  • 11A41
  • 11A55
  • 11A07

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Cite this paper

APA

Leyendekkers, J., & Shannon, A. (2011). The structure of ‘Pi’, Notes on Number Theory and Discrete Mathematics, 17(4), 61-68.

Chicago

Leyendekkers, JV, and AG Shannon. “The Structure of ‘Pi’.” Notes on Number Theory and Discrete Mathematics 17, no. 4 (2011): 61-68.

MLA

Leyendekkers, JV, and AG Shannon. “The Structure of ‘Pi’.” Notes on Number Theory and Discrete Mathematics 17.4 (2011): 61-68. Print.

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