On
Author:
Noam D. Elkies
Journal:
Math. Comp. 51 (1988), 825835
MSC:
Primary 11D25; Secondary 11G35, 11G40
MathSciNet review:
930224
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Abstract: We use elliptic curves to find infinitely many solutions to in coprime natural numbers A, B, C, and D, starting with We thus disprove the case of Euler's conjectured generalization of Fermat's Last Theorem. We further show that the corresponding rational points on the surface are dense in the real locus. We also discuss the smallest solution, found subsequently by Roger Frye.
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 A. Bremner, personal communication, Aug. 1987.
 [2]
 B. J. Birch & W. Kuyk, Editors, Modular Functions on One Variable IV, Lecture Notes in Math., vol. 476, SpringerVerlag, New York, 1975. MR 0376533 (51:12708)
 [3]
 V. A. Demjanenko, "L. Euler's conjecture," Acta Arith., v. 25, 197374, pp. 127135. (Russian) MR 0360462 (50:12912)
 [4]
 L. E. Dickson, History of the Theory of Numbers, Vol. II: Diophantine Analysis, G. E. Stechert & Co., New York, 1934.
 [5]
 R. K. Guy, Unsolved Problems in Number Theory, SpringerVerlag, New York, 1981. MR 656313 (83k:10002)
 [6]
 K. Ireland & M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, vol. 84, SpringerVerlag, New York, 1982. MR 661047 (83g:12001)
 [7]
 L. J. Lander & T. R. Parkin, "Counterexamples to Euler's conjecture on sums of like powers," Bull. Amer. Math. Soc., v. 72, 1966, p. 1079. MR 0197389 (33:5554)
 [8]
 B. Mazur, "Rational isogenies of prime degree," Invent. Math., v. 44, 1978, pp. 129162. MR 482230 (80h:14022)
 [9]
 J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106, SpringerVerlag, New York, 1986. MR 817210 (87g:11070)
 [10]
 D. Zagier, "On the equation ," unpublished note, 1987.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809302249
PII:
S 00255718(1988)09302249
Article copyright:
© Copyright 1988
American Mathematical Society
