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Almost powers in the Lucas sequence

Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek (2008)

Journal de Théorie des Nombres de Bordeaux

The famous problem of determining all perfect powers in the Fibonacci sequence ( F n ) n 0 and in the Lucas sequence ( L n ) n 0 has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations L n = q a y p , with a > 0 and p 2 , for all primes q < 1087 and indeed for all but 13 primes q < 10 6 . Here the strategy of [10] is not sufficient due to the sizes of...

Complete solutions of a family of cubic Thue equations

Alain Togbé (2006)

Journal de Théorie des Nombres de Bordeaux

In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation Φ n ( x , y ) = x 3 + ( n 8 + 2 n 6 - 3 n 5 + 3 n 4 - 4 n 3 + 5 n 2 - 3 n + 3 ) x 2 y - ( n 3 - 2 ) n 2 x y 2 - y 3 = ± 1 , for n 0 .

Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

István Gaál, Gábor Nyul (2001)

Journal de théorie des nombres de Bordeaux

Let M be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields K with mixed signature having power integral bases and containing M as a subfield. We also determine all generators of power integral bases in K . Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for M = ( 2 ) , ( 3 ) , ( 5 ) .

Effective results for Diophantine equations over finitely generated domains

Attila Bérczes, Jan-Hendrik Evertse, Kálmán Győry (2014)

Acta Arithmetica

Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over ℤ. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations f ( x ) = δ y m in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and m 2 . Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for A, δ, F, f, m. These...

Generators and integral points on twists of the Fermat cubic

Yasutsugu Fujita, Tadahisa Nara (2015)

Acta Arithmetica

We study integral points and generators on cubic twists of the Fermat cubic curve. The main results assert that integral points can be in a system of generators in the case where the Mordell-Weil rank is at most two. As a corollary, we explicitly describe the integral points on the curve.

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