J. V. Leyendekkers and A. G. Shannon

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

Volume 5, 1999, Number 2, Pages 71–79

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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

It has been suspected that if Fermat did indeed have a simple proof for his famous ‘last theorem’, that he probably employed his method of infinite descent. In a renewed attempt to see how Fermat might have thought that he had proved that if *c ^{n}* =

*a*+

^{n}*b*,

^{n}*a*,

*b*,

*c*,

*n*∈ ℤ,

*n*> 2, then

*a*,

*b*,

*c*cannot all be integers, we set

*c*=

*a*+

*b*+

*m*,

*m*∈ ℤ, and then raised it to the

*n-*th power. The roots of the resulting polynomial in

*m*appear to be – (

*a*+

*b*) only when

*n*≠ 1 ,2 , and the result might have seemed to Fermat to follow from this. The plausibility of the algebra developed here is considered in the context of the work of the sixteenth century mathematicians, particularly Cardano and Bombelli.

### AMS Classification

- 01A45
- 11D41

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## Cite this paper

Leyendekkers, J. V. & Shannon, A. G. (1999). Speculation of Fermat’s proof of his Last Theorem. *Notes on Number Theory and Discrete Mathematics*, 5(2), 71-79.