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Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
 Annals of Math
"... Abstract. This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general ..."
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Abstract. This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the LebesgueNagell equation for D in the range 1 ≤ D ≤ 100. x 2 + D = y n, x, y integers, n ≥ 3, 1.
Minimisation and Reduction of 2, 3 and 4coverings of elliptic curves
, 2009
"... In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit ndescent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms ..."
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Cited by 3 (1 self)
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In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit ndescent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general number fields. Finally, for genus one models defined over Q, we develop a theory of reduction and again give explicit algorithms for n = 2, 3 and 4.
Computing All Integer Solutions of a Genus 1 Equation
"... The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 defining elliptic curves. We extend this method to include any equation f(u, v) = 0, where f Z[u, v] is irreducible over Q, defines a curve of genus 1 ..."
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The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 defining elliptic curves. We extend this method to include any equation f(u, v) = 0, where f Z[u, v] is irreducible over Q, defines a curve of genus 1, but is otherwise of arbitrary shape and degree. We give a detailed description of the general features of our approach, and conclude with two rather unusual examples corresponding to equations of degree 5 and degree 9. 1991 Mathematics subject classification: 11D41, 11G05 Key words and phrases: diophantine equation, elliptic curve, elliptic logarithm # Econometric Institute, Erasmus University, P.O.Box 1738, 3000 DR Rotterdam, The Netherlands; email: stroeker@few.eur.nl; homepage: http://www.few.eur.nl/few/people/stroeker/ + Department of Mathematics, University of Crete, Iraklion, Greece; email: tzanakis@math.uch.gr; homepage: http://www.math.uoc.gr/tzanakis 1
Computing All Integer Solutions of a General Elliptic Equation
, 2000
"... The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 dening elliptic curves. We explore the possibility of extending this method to include any equation f(u; v) = 0, where f 2 Z[u;v] denes an irreducible cu ..."
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The Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 dening elliptic curves. We explore the possibility of extending this method to include any equation f(u; v) = 0, where f 2 Z[u;v] denes an irreducible curve of genus 1, independent of shape or degree of the polynomial f . We give a detailed description of the general features of our approach, putting forward along the way some claims (one of which conjectural) that are supported by the explicit examples added at the end. 1
THE BRAUERMANIN OBSTRUCTION FOR INTEGRAL POINTS ON CURVES
"... We discuss the question of whether the BrauerManin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations ..."
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We discuss the question of whether the BrauerManin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations and relate it to an old conjecture of Skolem. We show that the case of elliptic curves minus at least three points reduces to the case of rational curves. Finally, we show that for elliptic curves minus one point the question has a negative answer.
SIMATH  a computer algebra system for number theoretic applications
"... this paper is to give a survey of the wide range of number theoretic applications of the computer algebra system SIMATH [42] ..."
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this paper is to give a survey of the wide range of number theoretic applications of the computer algebra system SIMATH [42]
A NEW ALGORITHM TO SEARCH FOR SMALL NONZERO x 3 − y 2  VALUES
"... Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = x 3 −y 2  values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in ..."
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Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = x 3 −y 2  values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in other cases, it seems difficult to achieve small absolute values. For a given k ̸ = 0, (1.1), known as Mordell’s equation, is an elliptic curve and has only finitely many solutions in integers by Siegel’s theorem. Therefore, for any nonzero k value, there are only finitely many solutions in x (which is hence bounded). There is a proven lower bound, due to A. Baker [1] and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of k  was found early by M. Hall [2, 7] by means of a parametric family of the form (1.2) f(t) = t 9 (t9 +6t 6 +15t 3 + 12), g(t) = t15 27 + t12 +4t9 +8t6 3 f 3 (t) − g2 (t) = − 3t6 +14t3+27
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I agree that this thesis shall be available in accordance with the regulations governing the University of Warwick theses. I agree that the summary of this thesis may be submitted for publication. I agree that the thesis may be photocopied (single copies for study purposes only). Theses with no restriction on photocopying will also be made available to the British Library for microfilming. The British Library may supply copies to individuals or libraries. subject to a statement from them that the copy is supplied for nonpublishing purposes. All copies supplied by the British Library will carry the following statement: “Attention is drawn to the fact that the copyright of this thesis rests with its author. This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the author’s written consent.”