Rešenje diofantske jednačine
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 ?
-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.
S-ganze Punkte auf elliptischen Kurven.
Sharp bounds for the number of solutions to simultaneous Pellian equations
Small solutions of cubic equations with prime variables in arithmetic progressions
Solution de l’équation
Solutions des mêmes questions
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...
Solutions of x³+y³+z³=nxyz
The diophantine equation (1) x³ + y³ + z³ = nxyz has only trivial solutions for three (probably) infinite sets of n-values and some other n-values ([7], Chs. 10, 15, [3], [2]). The main set is characterized by: n²+3n+9 is a prime number, n-3 contains no prime factor ≡ 1 (mod 3) and n ≠ - 1,5. Conversely, equation (1) is known to have non-trivial solutions for infinitely many n-values. These solutions were given either as "1 chains" ([7], Ch. 30, [4], [6]), as recursive...
Solutions to infinite families of complex cubic Thue equations.
Solutions to xyz = 1 and x + y + z = k in algebraic integers of small degree, II
Let k ∈ ℤ be such that is finite, where . We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.
Solving a family of Thue equations with an application to the equation x²-Dy⁴=1
Solving elliptic diophantine equations avoiding Thue equations and elliptic logarithms.
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations
Solving elliptic diophantine equations: the general cubic case
Solving the equation x3 - y2 = r in number fields.