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

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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.