J. V. Leyendekkers and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 7, 2001, Number 2, Pages 48—59

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## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney
NSW 2006, Australia*

A. G. Shannon

*Warrane College, The University of New South Wales, 1465, &
KvB Institute of Technology, North Sydney, 2060, Australia*

### Abstract

Cubes and squares are expanded in various ways stimulated by Fibonacci’s odd number triangle which is in turn extended to even powers. The class structure of the cubes within the modular ring ℤ_{4} is developed. This provides constraints for the various functions which help in solving polynomial and Diophantine equations, some simple examples of which are given.

### AMS Classification

- 11C08
- 11B39

### References

- Abramowitz, M & Stegun, TA. 1964. Handbook of Mathematical Functions. Washington, DC: National Bureau of Standards.
- Ando, Shiro. 1982. On a System of Diophantine Equations Concerning the Polygonal Numbers. The Fibonacci Quarterly, Vol. 20(4): 349-353.
- Hogben, L. 1950. Choice and Chance by Cardpack and Chessboard. Volume 1. New York: Chanticleer Press
- Hollingdale, Stuart. 1989. Makers a/Mathematics. London: Penguin.
- Lehmer, DH. 1941. Guide to Tables in the Theory of Numbers. Bulletin No. 105. Washington, DC: National Research Council.
- Leyendekkers, JV, Rybak, 1M & Shannon, AG. 1997. Analysis of Diophantine Properties Using Modular Rings with Four and Six Classes. Notes on Number Theory & Discrete Mathematics. VoI. 3(2): 61-74.
- Mason, John. 2001. Generalising ‘Sums of Cubes Equal to Squares of Sums’. Mathematical Gazette. Vol. 85(502): 50-58.
- Müller, Siguna. 2000. Some Remarks on Primality Testing Based on Lucas Sequences. Ninth International Conference on Fibonacci Numbers and Their Applications, Luxembourg, 17-22 July.
- Utz, WR. 1977. The Diophantine Equation (x
_{1}+x_{2}+ … + x_{n})^{2}= x_{1}^{3}+ x_{2}^{3}+ … + x_{n}^{3}. The Fibonacci Quarterly. VoI.15(1): 14-16. - Wieckowski, Andrzej. 1980. On Some Systems of Diophantine Equations Including the Algebraic Sum of Triangular Numbers. The Fibonacci Quarterly. Vol. 18(2): 165-170.

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## Cite this paper

APALeyendekkers, J., & Shannon, A. (2001). The decimal string of the golden ratio. Notes on Number Theory and Discrete Mathematics, 7(2), 48-59.

ChicagoLeyendekkers, JV, and AG Shannon. “The Decimal String of the Golden Ratio.” Notes on Number Theory and Discrete Mathematics 7, no. 2 (2001): 48-59.

MLALeyendekkers, JV, and AG Shannon. “The Decimal String of the Golden Ratio.” Notes on Number Theory and Discrete Mathematics 7.2 (2001): 48-59. Print.