On pairs of binary forms with given resultant or given semi-resultant.
On some arithmetical properties of weighted sums of S-units.
Norm form équations.
Correction to the paper "Effective finiteness theorems for polynomials with given discriminant and integral elements with given discriminant over finitely generated domains".
Effective finiteness theorems for polynomials with given discriminant and integral elements with given discriminant over finitely domains.
On the number of solutions of linear equations in units of an algebraic number field.
On discriminants and indices of integers of an algebraic number field.
On the irreducibility of neighbouring polynomials
On the irreducibility of a class of polynomials, IV
Upper bounds for the degrees of decomposable forms of given discriminant
1. Introduction. In our paper [5] a sharp upper bound was given for the degree of an arbitrary squarefree binary form F ∈ ℤ[X,Y] in terms of the absolute value of the discriminant of F. Further, all the binary forms were listed for which this bound cannot be improved. This upper estimate has been extended by Evertse and the author [3] to decomposable forms in n ≥ 2 variables. The bound obtained in [3] depends also on n and is best possible only for n = 2. The purpose of the present paper is to establish...
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...
On the diophantine equation
On the diophantine equation
P. 294, line 14: For “Satz 8” read “Satz 7”, and for “equation (10)” read “equation (13)”.
On the abc conjecture in algebraic number fields
On unit equations and decompsable form equations.
Thue-Mahler equations with a small number of solutions.
Integer solutions of a sequence of decomposable form inequalities
On prime factors of integers of the form (ab+1)(bc+1)(ca+1)
1. Introduction. For any integer n > 1 let P(n) denote the greatest prime factor of n. Győry, Sárközy and Stewart [5] conjectured that if a, b and c are pairwise distinct positive integers then (1) P((ab+1)(bc+1)(ca+1)) tends to infinity as max(a,b,c) → ∞. In this paper we confirm this conjecture in the special case when at least one of the numbers a, b, c, a/b, b/c, c/a has bounded prime factors. We prove our result in a quantitative form by showing that if is a finite set of triples (a,b,c)...
On the number of solutions of decomposable polynomial equations
On the numbers of solutions of weighted unit equations
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