On unit solutions of the equation xyz = x+y+z in the ring of integers of a quadratic field
Points entiers sur les courbes de genre 0
Polynomials dividing infinitely many quadrinomials or quintinomials
Power integral bases in orders of composite fields.
Power integral bases in the family of simplest quartic fields.
p-primary parts of unit traces and the p-adic regulator
Products of disjoint blocks of consecutive integers which are powers
The product of consecutive integers cannot be a power (after Erdős and Selfridge), but products of disjoint blocks of consecutive integers can be powers. Even if the blocks have a fixed length l ≥ 4 there are many solutions. We give the bound for the smallest solution and an estimate for the number of solutions below x.
Rational solutions of certain Diophantine equations involving norms
We present some results concerning the unirationality of the algebraic variety given by the equation , where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then is k-unirational. A similar result is proved for a broad family of quintic polynomials f satisfying...
Representing integers as linear combinations of S-units
Solutions of cubic equations in quadratic fields
Let K be any quadratic field with its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over ℚ, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r+s+t = rst = 1 in . This Diophantine equation gives an elliptic curve defined over ℚ with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields...
Some applications of decomposable form equations to resultant equations
1. Introduction. The purpose of this paper is to establish some general finiteness results (cf. Theorems 1 and 2) for resultant equations over an arbitrary finitely generated integral domain R over ℤ. Our Theorems 1 and 2 improve and generalize some results of Wirsing [25], Fujiwara [6], Schmidt [21] and Schlickewei [17] concerning resultant equations over ℤ. Theorems 1 and 2 are consequences of a finiteness result (cf. Theorem 3) on decomposable form equations over R. Some applications of Theorems...
Some diophantine equations with many solutions
Some meromorphic functions associated to the S-unit equation
Some remarks on the S-unit equation in function fields
Sur certains nombres complexes compris dans la formule.
Sur des fractions rationnelles particulières
Sur la résolution du système des équations et en nombres entiers
Sur les équations diophantiennes liées aux unités d'un corps de nombres algébriques fini [Book]
Sur les générateurs des ordres monogènes des corps de nombres algébriques.
Symmetric Diophantine systems