Pour tout $c\<2$, on calcule un rang $D\left(c\right)$ tel que tout entier algébrique $x$ de degré au moins $D\left(c\right)$ ait deux conjugués $x^{\text{'}},x^{\text{'}\text{'}}$ vérifiant $|x^{\text{'}}-x^{\text{'}\text{'}}|\ge c$. De plus, on donne une nouvelle preuve de l’égalité $D\left(\sqrt{3}\right)=2$.

We solve a problem of Bombieri, stated in connection with the «prime number theorem» for function fields.

The paper consists of two parts, both related to problems of Lubelski, but unrelated otherwise. Theorem 1 enumerates for a = 1,2 the finitely many positive integers D such that every odd positive integer L that divides x² +Dy² for (x,y) = 1 has the property that either L or $2^{a}L$ is properly represented by x²+Dy². Theorem 2 asserts the following property of finite extensions k of ℚ : if a polynomial f ∈ k[x] for almost all prime ideals of k has modulo at least v linear factors, counting multiplicities,...

Soit $K$ un corps de nombres. Dans ce travail nous calculons des majorants effectifs pour la taille des solutions en entiers algébriques de $K$ des équations, $Y^2=f\left(X\right)$, où $f\left(X\right)\in K\left[X\right]$ a au moins trois racines d’ordre impair, et $Y^m=f\left(X\right)$ où $m\ge 3$ et $f\left(X\right)\in K\left[X\right]$ a au moins deux racines d’ordre premier à $m$. On améliore ainsi les estimations connues ([2],[9]) pour les solutions de ces équations en entiers algébriques de $K$.

Let K be any quadratic field with ${}_K$ 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 ${}_K$. 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...

Consider the system $x^2-a{y}^2=b$, $P(x,y)=z^2$, where $P$ is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation $7X^2+Y^7=Z^2$ if $(X,Y)=(L_n,F_n)$ (or $(X,Y)=(F_n,L_n)$) where $\left\{F_n\right\}$ and $\left\{L_n\right\}$ represent the sequences of Fibonacci numbers and Lucas numbers respectively....

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...