Pellian sequence relationships among π, e, √2

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 2, Pages 58–62
Full paper (PDF, 143 Kb)

Details

Authors and affiliations

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

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

Abstract

The numerators and denominators of the convergents of the continued fractions of π, e and √2 are shown to be elements of second order recurrence sequences of the Pellian or Fibonacci variety which are related to Pythagorean triples (c² = b² + a², b > a). π and √2 have surprisingly similar structures except that √2 has primitive Pythagorean triples with c − b = 1 or b − a = 1, whereas π has c − b even and not constant and b − a not constant, although the right-end-digits are constant.

Keywords

• Integer structure analysis
• Modular rings
• Prime numbers
• Fibonacci numbers
• Infinite series
• Pell sequence
• Continued fractions
• Primitive Pythagorean triples
• Right-end-digits

AMS Classification

• 11A41
• 11A55
• 11A07

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