Fermat's Last Theorem
نویسنده
چکیده
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.
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Proof for the Beal conjecture and a new proof for Fermat's last theorem
The Beal Conjecture was formulated in 1997 and presented as a generalization of Fermat's Last Theorem, within the number theory ́s field. It states that, for X, Y, Z, n , n and n positive integers with n , n , n > 2, if X + Y = Z then X, Y, Z must have a common prime factor. This article presents the proof for the Beal Conjecture, obtained from the correspondences between the real solutions of t...
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IH. S. Vandiver, "Fermat's Last Theorem," Am. Math. Mlonthly, 53, 567-568, 1946. 2 H. S. Vandiver, "The Relation of Some Data Obtained from Rapid Computing Machines to the Theory of Cyclotomic Fields, these PROCEEDINGS, 40, 474-480, 1954. 3H. S. Vandiver, "A Theorem of Kummer's concerning the Second Factor of the Class Number of a Cyclotomic Field," Bull. Am. Math. Soc., 35, 333-335, 1929. 4D. ...
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This note continues work by the Lehmers [3], Gunderson [2], Granville and Monagan [1], and Tanner and Wagstaff [6], producing lower bounds for the prime exponent p in any counterexample to the first case of Fermat's Last Theorem. We improve the estimate of the number of residues r mod p" such that fP = r mod p~ , and thereby improve the lower bound on p to 7.568 x 1017.
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Fermat's Last Theorem—which we shall abbreviate to FLT—is the (as yet unproved) assertion that the Diophantine equation x" + y" — z" has no solutions in positive integers if 11 ^ 3. It would suffice to deal with the case in which n is prime, and this is where the most significant work has been done. None the less it is not yet known even whether FLT is true for infinitely many prime exponents. ...
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13 Ibid., ref. 11; "Examination of methods of attack on the second case of Fermat's last theorem," these PROCEEDINGS, 40, 732-735 (1954); Selfridge, J. L., C. A. Nicol, and V., "Proof of Fermat's last theorem for all prime exponents less than 4002," these PROCEEDINGS, 41, 970-973 (1955). 14 Sitz. Akad. Wiss. Wien, 121, Ila, 589-592 (1912). 11 In one of the author's previously published papers (...
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