On the diophantine equations x2 + 74 = y5 and x2 + 86 = y5

THE DIOPHANTINE EQUATION x2+2k =yn, II

New results regarding the full solution of the diophantine equationx2+2k=yn in positive integers are obtained. These support a previous conjecture, without providing a complete proof.

ON THE DIOPHANTINE EQUATION x2+p2k+1 = 4yn

On the System of Diophantine Equations x2 − 6y2 = −5 and x = az2 − b

Mignotte and Pethö used the Siegel-Baker method to find all the integral solutions (x, y, z) of the system of Diophantine equations x (2) - 6y (2) = -5 and x = 2z (2) - 1. In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equations x (2) - 6y (2) = -5 and x = az (2) - b for each pair of integral parameters a...

Solving Fermat-type equations x5+y5=dzp

In this paper, we are interested in solving the Fermat-type equations x+y = dz where d is a positive integer and p a prime number ≥ 7. We describe a new method based on modularity theorems which allows us to improve all the results of [1]. We finally discuss the present limitations of the method by looking at the case d = 3.

ON THE EQUATION x2+2a ·3b =yn