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3-Selmer groups for curves y 2 = x 3 + a

Andrea Bandini (2008)

Czechoslovak Mathematical Journal

We explicitly perform some steps of a 3-descent algorithm for the curves y 2 = x 3 + a , a a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.

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...

Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields

István Gaál, Gábor Petrányi (2014)

Czechoslovak Mathematical Journal

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields K generated by a root ξ of the polynomial P t ( x ) = x 4 - t x 3 - 6 x 2 + t x + 1 , assuming that t > 0 , t ≠ 3 and t 2 + 16 has no odd square factors. In addition to generators of power integral bases we also calculate the minimal...

Catalan’s conjecture

Yuri F. Bilu (2002/2003)

Séminaire Bourbaki

The subject of the talk is the recent work of Mihăilescu, who proved that the equation x p - y q = 1 has no solutions in non-zero integers x , y and odd primes p , q . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebratedconjecture of Catalan (1843): the only solution to x u - y v = 1 in integers x , y > 0 and u , v > 1 is 3 2 - 2 3 = 1 . Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute...

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 = ℚ ( 2 ) , ℚ ( 3 ) , ℚ ( 5 ) .

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