Speculation of Fermat’s proof of his Last Theorem

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
Full paper (PDF, 328 Kb)

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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 cⁿ = aⁿ + bⁿ, 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

References

1. Boyer Carl B. 1968. A History; of Mathematics. Princeton: Princeton University Press.
2. Cajori F. 1928. A History of Mathematical Notations. Volumes 1,2. La Salle: Open Court.
3. Cohen G L & Shannon AG. 1981. John Ward’s method for the calculation of pi. Historia Mathematica. 8: 133-144.
4. Franklin, James & Daoud, Albert. 1988 Introduction to Proofs in Mathematics. Sydney: Prentice Hall.
5. Hillman, Abraham P & Alexanderson Gerald P 1978. A First Undergraduate Course in Abstract Algebra Second Edition. Belmont, CA: Wadsworth.
6. Hollingdale, Stewart. 1989. Makers of Mathematics. London. Penguin.
7. Hunter, J 1964. Number Theory. Edinburgh: Oliver and Boyd.
8. Petsinis, Tom. 1997. The French Mathematician Ringwood, Vic.: Penguin.
9. Poorten, Alf van der. 1996. Notes on Fermat’s Last Theorem New York: John Wiley.
10. Riemer, Andrew. 1998. Sandstone Gothic. Sydney: Allen and Unwin.
11. Singh, Simon & Ribet, Kenneth A. 1997. Fermat’s Last Stand Scientific American. 277.5: 36-41 Stillwell, John. 1998. Numbers and Geometry. New York: Springer.
12. Stillwell, Dirk J. 1965. A Concise History of Mathematics. London. George Bell.
13. Velleman, Daniel J. 1997. Fermat’s Last Theorem and Hilbert’s Program. Mathematical Intelligencer. 19.1 64-67.