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Some Diophantine equations in three variables with only finitely many solutions

Published online by Cambridge University Press:  26 February 2010

Wolfgang M. Schmidt
Wolfgang M. Schmidt
Affiliation:
University of Colorado, Boulder, Colorado, U.S.A.
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Extract

This paper deals with diophantine equations and inequalities in three variables x, y, z. Write

Given an algebraic number field K of degree n and an element ξ of K, denote the conjugates of ξ by ξ(1) = ξ, ξ(2), …, ξ(n), and its norm ξ(1) … ξ(n) by N(ξ). Recent results [3] in diophantine approximations enable us to prove the following theorems.

Type
Research Article
Information
Mathematika , Volume 14 , Issue 2 , December 1967 , pp. 113 - 120
Copyright
Copyright © University College London 1967

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References

1.Chabauty, C., “Sur certaines équations diophantiques ternaires”, C.R. Acad. Sci. Paris, 202 (1936), 2117–2119.Google Scholar
2.Roth, K. F., “Rational approximations to algebraic numbers”, Mathematika, 2 (1955), 1–20.Google Scholar
3.Schmidt, W. M., “On simultaneous approximations to two algebraic numbers”, Acta Mathematica. (To appear.)Google Scholar
4.Skolem, Th., “Einige Sätze über p-adische Potenzreihen mit Anwendung auf gewisse exponentielle Gleichungen”, Math. Annalen, 111 (1935), 399–424.CrossRefGoogle Scholar