Fermat's Last Theorem

About: Fermat's Last Theorem is a research topic. Over the lifetime, 1223 publications have been published within this topic receiving 16427 citations. The topic is also known as: Fermat's conjecture & Wiles' theorem.

Modular elliptic curves and Fermat’s Last Theorem

Abstract: When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

Elementary Number Theory

Abstract: 1. Some Preliminary Considerations. 2. Divisibility Theory in the Integers. 3. Primes and Their Distribution. 4. The Theory of Congruences. 5. Fermat's Theorem. 6. Number-Theoretic Functions. 7. Euler's Generalization of Fermat's Theorem. 8. Primitive Roots and Indices. 9. The Quadratic Reciprocity Law. 10. Perfect Numbers. 11. The Fermat Conjecture. 12. Representation of Integers as Sums of Squares. 13. Fibonacci Numbers. 14. Continued Fractions. 15. Some Twentieth-Century Developments.

Fermat’s Last Theorem

Abstract: The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this paper. They are also grateful to A. Agboola, M. Bertolini, B. Edixhoven, J. Fearnley, R. Gross, L. Guo, F. Jarvis, H. Kisilevsky, E. Liverance, J. Manoharmayum, K. Ribet, D. Rohrlich, M. Rosen, R. Schoof, J.-P. Serre, C. Skinner, D. Thakur, J. Tilouine, J. Tunnell, A. Van der Poorten, and L. Washington for their helpful comments. Darmon thanks the members of CICMA and of the Quebec-Vermont Number Theory Seminar for many stimulating conversations on the topics of this paper, particularly in the Spring of 1995. For the same reason Diamond is grateful to the participants in an informal seminar at Columbia University in 1993-94, and Taylor thanks those attending the Oxford Number Theory Seminar in the Fall of 1995.

A Generalization of a Theorem of Baker and Davenport

Abstract: The Greek mathematician Diophantus of Alexandria noted that the rational numbers 1 16 , 33 16 , 17 4 and 105 16 have the following property: the product of any two of them increased by 1 is a square of a rational number (see [4]). The first set of four positive integers with the above property was found by Fermat, and it was {1, 3, 8, 120}. A set of positive integers {a1, a2, . . . , am} is said to have the property of Diophantus if aiaj +1 is a perfect square for all 1 ≤ i < j ≤ m. Such a set is called a Diophantine m-tuple (or P1-set of size m). In 1969, Baker and Davenport [2] proved that if d is a positive integer such that {1, 3, 8, d} is a Diophantine quadruple, then d has to be 120. The same result was proved by Kanagasabapathy and Ponnudurai [9], Sansone [12] and Grinstead [7]. This result implies that the Diophantine triple {1, 3, 8} cannot be extended to a Diophantine quintuple. In the present paper we generalize the result of Baker and Davenport and prove that the Diophantine pair {1, 3} can be extended to infinitely many Diophantine quadruples, but it cannot be extended to a Diophantine quintuple.