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We prove that for n ? 30, every n-th 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.

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.

In this paper we prove that the Diophantine equation where c 3 is an integer, has only the trivial solutions (1, 0), (0, 1).

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, e-mail: 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...

In this paper we prove that the only primitive solutions of the 1). 1

The complete sets of solutions of the equation ( n) ( m) = are k ℓ

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 Thue-Siegel 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.

ABSTRACT. In an earlier paper we determined all the solutions in Z of a cubic Thue equation with coefficients in a quadratic number field. It is now shown that the method used there can be used to solve the more general problem of determining all the solutions of the Thue equation in a ring of quadratic integers. 1.

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