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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 Lebesgue-Nagell equation for D in the range 1 ≤ D ≤ 100. x 2 + D = y n, x, y integers, n ≥ 3, 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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Cited by 3 (3 self)
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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
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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Cited by 3 (0 self)
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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; e-mail: stroeker@few.eur.nl; homepage: http://www.few.eur.nl/few/people/stroeker/ + Department of Mathematics, University of Crete, Iraklion, Greece; e-mail: tzanakis@math.uch.gr; homepage: http://www.math.uoc.gr/tzanakis 1
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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Cited by 1 (0 self)
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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]
THE BRAUER-MANIN OBSTRUCTION FOR INTEGRAL POINTS ON CURVES
"... We discuss the question of whether the Brauer-Manin 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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Cited by 1 (1 self)
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We discuss the question of whether the Brauer-Manin 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.
Basic Algorithms for Elliptic Curves
"... b 2 b 4 216b 6 : The discriminant is then = b 2 2 b 8 8b 3 4 27b 2 6 + 9b 2 b 4 b 6 and the modular invariant j = c 3 4 : However, if E is dened over a number eld K, we shall use the short Weierstrass form (1:2) E : Y 2 = X 3 + aX + b (a; b 2 K) with discriminant = 16(4a 3 + 27b ..."
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b 2 b 4 216b 6 : The discriminant is then = b 2 2 b 8 8b 3 4 27b 2 6 + 9b 2 b 4 b 6 and the modular invariant j = c 3 4 : However, if E is dened over a number eld K, we shall use the short Weierstrass form (1:2) E : Y 2 = X 3 + aX + b (a; b 2 K) with discriminant = 16(4a 3 + 27b 2 ) = 16 0 and modular invariant j = 12 3 4a 3<F12.2
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
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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
NOTE ON J.H.E. COHN’S PAPER ”THE DIOPHANTINE EQUATION x n = Dy 2 + 1”
"... A classical problem of the theory of diophantine equations is to find the solutions of (1) x n = Dy 2 + 1 ..."
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A classical problem of the theory of diophantine equations is to find the solutions of (1) x n = Dy 2 + 1
Minimisation and Reduction of . . .
, 2009
"... In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit n-descent 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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In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit n-descent 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.
The Diophantine Equation b²X^4 - dY² = 1
, 1999
"... If b and d are given positive integers with b>1, then we show that the equation of the title possesses at most one solution in positive integers X, Y. Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The p ..."
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If b and d are given positive integers with b>1, then we show that the equation of the title possesses at most one solution in positive integers X, Y. Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.
