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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## 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 (*c*^{2} = *b*^{2} + *a*^{2}, *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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## Cite this paper

APALeyendekkers, J. V., & Shannon, A. (2012). Pellian sequence relationships among π, e, √2, Notes on Number Theory and Discrete Mathematics, 18(2), 58-62.

ChicagoLeyendekkers, JV, and AG Shannon. “Pellian Sequence Relationships among π, e, √2” Notes on Number Theory and Discrete Mathematics, 18, no. 2 (2012): 58-62.

MLALeyendekkers, Tieling, and AG Shannon. “Pellian Sequence Relationships among π, e, √2” Notes on Number Theory and Discrete Mathematics, 18.2 (2012): 58-62. Print.