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
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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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