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
Download 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
References
- Ball, W.W.R., H.S.M. Coxeter. 1956. Mathematical Recreations and Essays. New York: Macmillan.
- Bennett, A.A. 1925. Two New Arctangent Relations for π. The American Mathematical Monthly. 32: 253–255.
- Birch, R.H. 1946. An Algorithm for the Construction of Arctangent Relations. Journal of the London Mathematical Society. 21: 173–174.
- Horadam, A.F. 1971. Pell Identities. The Fibonacci Quarterly. 9: 245–252.
- Kasner, E., J. Newman. 1959. Mathematics and the Imagination. Norwich: Jarrold.
- Lafer, Phil. 1971. Discovering the Square-triangular Numbers. The Fibonacci Quarterly. 9: 93-105.
- Lehmer, D.H. 1936. Problem 3801. The American Mathematical Monthly. 43: 580.
- Lehmer, D.H. 1938. On Arctangent Relations for π. The American Mathematical Monthly. 45: 657–664.
- Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No 9.
- Leyendekkers, J.V., A.G. Shannon. 2011. Modular Rings and the Integer 3. Notes on Number Theory & Discrete Mathematics. 17 (2): 47–51.
- Leyendekkers, J.V., A.G. Shannon. 2011. The Modular Ring Z5. Notes on Number Theory & Discrete Mathematics. In press.
- Mack, J.M. 1970. The Continued Fraction Algorithm. Bulletin of the Australian Mathematical Society. 3: 413–422.
- Nimbran, Amrik Singh. 2010. On the Derivation of Machin-like Arctangent Identities for Computing Pi (π). The Mathematics Student. 79: 171–186.
- Shanks, D., J.W. Wrench Jr. 1962. Calculation of π to 100,000 Decimals. Mathematics of Computation. 16: 76–99.
- Sierpinski, W. 1964. Sur les Nombres Pentagonaux. Bulletin de la Société Royale des Sciences de Liège. 33: 513–517.
- Stormer, C. 1899. Solution complet en nombres entiers de l’equation m.arctan g[1/x] + n.arctan g[1/y] = k[π/4]. Bulletin de la Société Mathématique de France. 27: 160–170.
- Todd, John. 1949. A Problem on Arc Tangent Relations. The American Mathematical Monthly. 56: 517–528.
- Wetherfield, Michael Roby. 1996. The Enhancement of Machin’s Formula by Todd’s Process. The Mathematical Gazette. 80: 333–344.
Related papers
Cite this paper
Leyendekkers, J., & Shannon, A. (2011). The structure of ‘Pi’, Notes on Number Theory and Discrete Mathematics, 17(4), 61-68.