Diophantine representation of generalized Fibonacci numbers.
Diophantine representation of Mersenne and Fermat primes
Diophantine triples with values in binary recurrences
In this paper, we study triples and of distinct positive integers such that and are all three members of the same binary recurrence sequence.
Diophantische Analysis und Modulfunktionen.
Distribution of irregular prime numbers.
Distribution of Mordell-Weil ranks of families of elliptic curves
We discuss the distribution of Mordell-Weil ranks of the family of elliptic curves y² = (x + αf²)(x + βbg²)(x + γh²) where f,g,h are coprime polynomials that parametrize the projective smooth conic a² + b² = c² and α,β,γ are elements from ℚ̅. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.
Distribution of solutions of diophantine equations , where are polynomials
Distribution of solutions of diophantine equations , where are polynomials. - II
Divisibility sequences and powers of algebraic integers.
Division-ample sets and the Diophantine problem for rings of integers
We prove that Hilbert’s Tenth Problem for a ring of integers in a number field has a negative answer if satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over ). We relate division-ample sets to arithmetic of abelian varieties.
Du premier cas du theoreme de Fermat
Effective bounds for the zeros of linear recurrences in function fields
In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable.Moreover, we study similar problems in this context as the equation , where is a linear recurring sequence of polynomials and is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].
Effective diophantine approximation on
Effective estimates of solutions of some diophantine equations
Effective finiteness results for binary forms with given discriminant
Effective finiteness theorems for decomposable forms of given discriminant
Effective results for Diophantine equations over finitely generated domains
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 in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and . 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...
Effective results for linear equations in two unknowns from a multiplicative division group
Effective solution of families of Thue equations containing several parameters
Effective solution of the D(-1)-quadruple conjecture