Representations of integers by binary forms and the rank of the Mordell-Weil group.
Representations of multivariate polynomials by sums of univariate polynomials in linear forms
The paper is concentrated on two issues: presentation of a multivariate polynomial over a field K, not necessarily algebraically closed, as a sum of univariate polynomials in linear forms defined over K, and presentation of a form, in particular a zero form, as the sum of powers of linear forms projectively distinct defined over an algebraically closed field. An upper bound on the number of summands in presentations of all (not only generic) polynomials and forms of a given number of variables and...
Representing integers as linear combinations of S-units
Řešení diofantických rovnic rozkladem nad číselnými tělesy
Článek představuje zjednodušené základy teorie kvadratických zbytků a algebraické teorie čísel a jejich užití při řešení diofantických rovnic. Obsahuje i několik příkladů pro čtenáře.
Řešení rovnic neurčitých o více než dvou neznámých použitím řetězců
Řešení rovnice celými čísly
Rešenje diofantske jednačine
Résidus de puissances
Résolution de l’équation indéterminée en nombres entiers.
Résolution, en nombres entiers et sous sa forme la plus générale, de l'équation cubique, homogène, à trois inconnues
Résultats élémentaires sur certaines équations diophantiennes
Dans des travaux profonds, W. Ljunggren a montré que, pour donné, les équations diophantiennes and ont au plus ou solutions non triviales. Par des méthodes élémentaires, je réponds ici à la question : pour quelles valeurs de , premières ou analogues, ont-elles des solutions non-triviales ?
Rezolution en x de la Congruence x^n+my^n==0 (mod z^n)
-integral points on elliptic curves - Notes on a paper of B. M. M. de Weger
In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and -adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.
-integral solutions to a Weierstrass equation
The rational solutions with as denominators powers of to the elliptic diophantine equation are determined. An idea of Yuri Bilu is applied, which avoids Thue and Thue-Mahler equations, and deduces four-term (-) unit equations with special properties, that are solved by linear forms in real and -adic logarithms.
Searching for Diophantine quintuples
We consider Diophantine quintuples a, b, c, d, e. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most Diophantine quintuples.
Sets with even partition functions and 2-adic integers. II.
S-ganze Punkte auf elliptischen Kurven.
Sharp bounds for the number of solutions to simultaneous Pellian equations
Siegel’s theorem and the Shafarevich conjecture
It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field and any finite set of places of , one can effectively compute the set of isomorphism classes of hyperelliptic curves over with good reduction outside . We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus would imply an effective version of Siegel’s theorem for integral points on...
Simple groups of square order and interesting sequence of primes