Results 1  10
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31
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
 ACTA ARITHMETICA
, 1994
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Existence of primitive divisors of Lucas and Lehmer numbers
 J. Reine Angew. Math
, 2001
"... We prove that for n ? 30, every nth Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay. ..."
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Cited by 35 (0 self)
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We prove that for n ? 30, every nth Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay.
A parametric family of quartic Thue equations
, 2002
"... In this paper we prove that the Diophantine equation where c 3 is an integer, has only the trivial solutions (1, 0), (0, 1). ..."
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Cited by 13 (4 self)
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In this paper we prove that the Diophantine equation where c 3 is an integer, has only the trivial solutions (1, 0), (0, 1).
A family of quartic Thue inequalities
 Acta Arith
, 2004
"... In this paper we prove that the only primitive solutions of the 1). 1 ..."
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Cited by 12 (6 self)
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In this paper we prove that the only primitive solutions of the 1). 1
Solving Thue equations without the full unit group
 Math. Comp
"... Abstract. The main problem when solving a Thue equation is the computation of the unit group of a certain number field. In this paper we show that the knowledge of a subgroup of finite index is actually sufficient. Two examples linked with the primitive divisor problem for Lucas and Lehmer sequences ..."
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Cited by 10 (1 self)
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Abstract. The main problem when solving a Thue equation is the computation of the unit group of a certain number field. In this paper we show that the knowledge of a subgroup of finite index is actually sufficient. Two examples linked with the primitive divisor problem for Lucas and Lehmer sequences are given. 1.
Solving Elliptic Diophantine Equations: The General Cubic Case
 Acta Arith
, 1999
"... In this paper we consider binary cubic diophantine equations of every form and shape, solely subjected to the requirement that they represent elliptic curves defined over Q . How to deal with standard Weierstraß equations is well understood, but comparatively little is known about elliptic equations ..."
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Cited by 8 (5 self)
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In this paper we consider binary cubic diophantine equations of every form and shape, solely subjected to the requirement that they represent elliptic curves defined over Q . How to deal with standard Weierstraß equations is well understood, but comparatively little is known about elliptic equations of different type. We give a detailed analysis of the general situation and subsequently apply the elliptic logarithm method to a variety of unusual elliptic equations, notably to some equations directly related to Krawtchouk polynomials. 1991 Mathematics subject classification: 11D25, 11G05, 11Y50, 12D10 Key words and phrases: cubic diophantine equation, elliptic curve, elliptic logarithm, LLLreduction, binary Krawtchouk polynomial Econometric Institute, Erasmus University, P.O.Box 1738, 3000 DR Rotterdam, The Netherlands, email: stroeker@few.eur.nl, internet URL of my homepage: http://www.few.eur.nl/few/people/stroeker/ y Sportsingel 30, 2924 XN Krimpen aan den IJssel, The Netherla...
On the representation of unity by binary cubic forms
 Trans. Amer. Math. Soc
"... Abstract. If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) =1 possesses at most ten solutions in integers x and y, nineifF has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z ..."
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Abstract. If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) =1 possesses at most ten solutions in integers x and y, nineifF has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z[x, y], then this equation has at most 2 solutions, unless F (x, y) is equivalent under GL2(Z)action to either x(x 2 − xy − y 2)orx(x 2 − 2y 2). The proofs of these results rely upon the method of ThueSiegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations F (x, y) =1forF cubic and irreducible of positive discriminant DF ≤ 10 6. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form F (x, y) =m and to Mordell’s equation y 2 = x 3 + k, wherem and k are nonzero integers. 1.
ELLIPTIC BINOMIAL DIOPHANTINE EQUATIONS
, 1999
"... The complete sets of solutions of the equation ( n) ( m) = are k ℓ ..."
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Cited by 6 (4 self)
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The complete sets of solutions of the equation ( n) ( m) = are k ℓ
On a Conjecture of E. Thomas concerning parametrized Thue Equations
, 2000
"... this paper, we shall prove Thomas' conjecture for a very large class of polynomials p i , subject only to certain technical conditions on the degrees of the p i . After the presentation of the results in this section we shall, in Section 2, give some comments on the technical hypothesis occuring in ..."
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Cited by 5 (1 self)
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this paper, we shall prove Thomas' conjecture for a very large class of polynomials p i , subject only to certain technical conditions on the degrees of the p i . After the presentation of the results in this section we shall, in Section 2, give some comments on the technical hypothesis occuring in our Theorems 1 and 2 and we will discuss its relation to Thomas' technical hypothesis in the case n = 3. In Sections 3 and 4 we collect and adapt standard material for the solution of parametrized Thue equations. In Section 5 we will present the main idea to exclude \small" solutions, which will be carried out in detail in Sections 6 and 7. In Section 8 we will exclude \large" solutions using Baker's method via application of a result of Bugeaud and Gy}ory [5]. Finally we will prove a weaker formulation of the technical hypothesis in Section 9. The main result of the present paper is