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31
On the practical solution of the Thue equation
 INSTITUTE OF MATHEMATICS, UNIVERSITY OF DEBRECEN
, 1989
"... This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker’s theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented. ..."
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Cited by 48 (13 self)
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This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker’s theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented.
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 41 (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 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 16 (8 self)
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In this paper we prove that the only primitive solutions of the 1). 1
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).
Algebraic Geometry Over Four Rings and the Frontier to Tractability
 CONTEMPORARY MATHEMATICS
"... We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set (b) the height of the zerodimensional part of an algeb ..."
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Cited by 8 (4 self)
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We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set (b) the height of the zerodimensional part of an algebraic set over C (c) the number of connected components of a semialgebraic set We also present some results which significantly lower the complexity of deciding the emptiness of hypersurface intersections over C and Q, given the truth of the Generalized Riemann Hypothesis. Furthermore, we state some recent progress on the decidability of the prefixes 989 and 9989, quantified over the positive integers. As an application, we conclude with a result connecting Hilbert's Tenth Problem in three variables and height bounds for integral points on algebraic curves. This paper
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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Cited by 8 (0 self)
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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.
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 occurin ..."
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Cited by 6 (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
A system of relative Pellian equations and a related family of relative Thue equations, Preprint available at http://www.finanz.math.tugraz.at/~ziegler/Publications/PellEqV7.pdf
"... In this paper we consider the family of systems (2c + 1)U 2 − 2cV 2 = µ and (c − 2)U 2 − cZ 2 = −2µ of relative Pellian equations, where the parameter c and the root of unity µ are integers in the same imaginary quadratic number field K = Q √ −D. We show that only for c > 4 certain values of µ ..."
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Cited by 3 (0 self)
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In this paper we consider the family of systems (2c + 1)U 2 − 2cV 2 = µ and (c − 2)U 2 − cZ 2 = −2µ of relative Pellian equations, where the parameter c and the root of unity µ are integers in the same imaginary quadratic number field K = Q √ −D. We show that only for c > 4 certain values of µ yield solutions of this system, and solve the system completely for c  ≥ 1544686. Furthermore we will consider the related relative Thue equation X 4 − 4cX 3 Y + (6c + 2)X 2 Y 2 + 4cXY 3 + Y 4 = µ
Parametrized Thue Equations — A Survey
, 2005
"... We consider families of parametrized Thue equations Fa(X, Y) = ±1, a ∈ where Fa ¡ ∈ [a][X,Y] is a binary irreducible form with coefficients which are polynomials in some parameter a. We give a survey on known results. 1 Thue Equations Let F ∈ Z[X, Y] be a homogeneous, irreducible polynomial of deg ..."
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We consider families of parametrized Thue equations Fa(X, Y) = ±1, a ∈ where Fa ¡ ∈ [a][X,Y] is a binary irreducible form with coefficients which are polynomials in some parameter a. We give a survey on known results. 1 Thue Equations Let F ∈ Z[X, Y] be a homogeneous, irreducible polynomial of degree n ≥ 3 and m be a nonzero integer. Then the Diophantine equation F (X, Y) = m (1) is called a Thue equation in honour of A. Thue, who proved in 1909 [57]: Theorem 1 (Thue). (1) has only a finite number of solutions (x, y) ∈ Z 2. Thue’s proof is based on his approximation theorem: Let α be an algebraic number of degree n ≥ 2 and ɛ> 0. Then there exists a constant c1(α, ɛ), such that for all p ∈ Z and q ∈ N
THE METHOD OF THUESIEGEL FOR BINARY QUARTIC FORMS
, 906
"... Abstract. We will use ThueSiegel method, based on Padé approximation via hypergeometric functions, to give upper bounds for the number of integral solutions to the equation F(x, y)  = 1 as well as the inequalities F(x, y)  ≤ h, for a certain family of irreducible quartic binary forms. 1. ..."
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Cited by 1 (0 self)
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Abstract. We will use ThueSiegel method, based on Padé approximation via hypergeometric functions, to give upper bounds for the number of integral solutions to the equation F(x, y)  = 1 as well as the inequalities F(x, y)  ≤ h, for a certain family of irreducible quartic binary forms. 1.