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
Full paper (PDF, 119 Kb)

Details

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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