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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