Journal ArticleDOI
Integer points on curves of genus 1
A. Baker,John Coates +1 more
- Vol. 67, Iss: 3, pp 595-602
TLDR
In this paper, it was shown that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve of genus ≥ 1, and that such an algorithm cannot be found in the case of curves of genus 1.Abstract:
1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.read more
Citations
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The Arithmetic of Elliptic Curves
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There are only finitely many Diophantine quintuples
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Über einige Anwendungen diophantischer Approximationen
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Contributions to the Theory of Diophantine Equations. I. On the Representation of Integers by Binary Forms
TL;DR: In this paper, an effective algorithm was established for solving in integers x, y any Diophantine equation of the type/(x, y ) = m, where/ denotes an irreducible binary form with integer coefficients and degree at least 3.
Journal ArticleDOI
Bounds for the solutions of the hyperelliptic equation
TL;DR: In this paper, the authors extend the result which they established on the Diophantine equation to some further equations of a similar kind, and prove the following theorems for these equations.