J. V. Leyendekkers and A. G. Shannon

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

Volume 5, 1999, Number 4, Pages 151–162

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## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney, 2006, Australia*

A. G. Shannon

*University of Technology, Sydney, 2007, Australia*

### Abstract

The polynomial expansion of the Diophantine equation , yields roots of the form where is a non-integer zero of a Cardano cubic polynomial of the form . This is a corollary to Fermat’s Last Theorem. The characteristics of this family are illustrated for . For odd, can be represented by , and for even there are two real values of , and , where are real non-integer parameters. For a given , is simply related to , and to a parameter which is linear in . The corresponding curves indicate the non-integral nature of .

### AMS Classification

- 11C08
- 11D41

### References

- Boyer, Carl B. 1985. A History of Mathematics. Princeton: Princeton University Press.
- Dunham, William. 1990. Journey through Genius: The Great Theorems of Mathematics. New York: John Wiley.
- Galbraith, Steven. 1999. Elliptic Curve Public Key Cryptography. Mathematics Today’. 35(3): 76-79.
- Gould, Henry’ W. 1999. The Girard – Waring Power Sum Formulas for Symmetric Functions and Fibonacci Sequences. The Fibonacci Quarterly. 37(2): 135-140.
- Herz-Fischler, Roger. 1998. A Mathematical History of the Golden Section. New York: Dover.
- Hillman, Abraham P. & Alexanderson, Gerald L. 1978. A First Undergraduate Course in Abstract Algebra. Second Edition. Belmont, CA: Wadsworth.
- Householder, A.S. 1970. The Numerical Treatment of a Single Non-linear Equation. New York: McGraw-Hill.
- McLeish, John. 1991. The Story of Numbers. New York: Fawcett Columbine.
- Macmahon, Percy A. 1915. Combinatory Analysis. Volume I. Cambridge: Cambridge University Press.
- de Pillis, L.G. 1998. Newton’s Cubic Roots. The Australian Mathematical Society Gazette. 25(5): 236-241.
- Turnbull, H.W. 1952. Theory of Equations. Fifth Edition. Edinburgh: Oliver and Boyd.
- van der Poorten, A. 1996. Notes on Fermat’s Last Theorem. New York: Wiley.

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## Cite this paper

Leyendekkers, J. V. & Shannon, A. G. (1999). The Cardano family of equations. *Notes on Number Theory and Discrete Mathematics*, 5(4), 151-162.