Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T18:32:39.605Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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