This paper studies integer solutions to the $abc$ equation $A+B+C=0$ in which none of $A,B,C$ have a large prime factor. We set $H(A,B,C)=max\left(\right|A|,|B|,|C\left|\right)$, and consider primitive solutions ($\gcd (A,B,C)=1$) having no prime factor larger than ${(logH(A,B,C))}^{\kappa }$, for a given finite $\kappa$. We show that the $abc$ Conjecture implies that for any fixed $\kappa \<1$ the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed $\kappa \>8$ the $abc$ equation has infinitely many primitive solutions....

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...

Let k ∈ ℤ be such that ${|}_k\left(\mathbb{Q}\right)|$ is finite, where ${}_k:y²=1-2kx+k²x²-4x³$. 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.

Let k ∈ ℤ be such that ${|}_k\left(\mathbb{Q}\right)|=3$, where ${}_k:y²=1-2kx+k²x²-4x³$. We determine all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.