On Thue's equation.
On unit equations and decompsable form equations.
On Wendt's determinant and Sophie Germain's theorem.
On and
The two related Diophantine equations: and , have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.
On
In p. 219 of R.K. Guy’s Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation has no integer solutions with and . But, contrary to this expectation, we show that for , this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition .
Osservazioni sul problema di Fermat.
Pairs of additive equations IV. Sextic equations
Parametric solutions for some Diophantine equations.
Perfect powers expressible as sums of two fifth or seventh powers
We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the non-existence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.
Perfect powers in arithmetic progression. A note on the inhomogeneous case
Perfect powers in products of integers from a block of consecutive integers
Points d'ordre fini des variétés abéliennes de dimension un
Points entiers sur les courbes hyperelliptiques
Points rationnels et méthode de Chabauty elliptique
La méthode de Chabauty elliptique permet de calculer les points rationnels sur une courbe définie sur un corps de nombres lorsque le théorème de Chabauty ne s’applique pas, c’est à dire lorsque le rang de la jacobienne est supérieur au genre de la courbe. Nous exposons cette méthode et nous la généralisons dans de nouveaux cas en écrivant une version explicite du théorème de préparation de Weierstrass en variables. En particulier nous calculons tous les points rationnels d’une courbe de genre...
Polynomial squares of the form
Power values of sums of products of consecutive integers
We investigate power values of sums of products of consecutive integers. We give general finiteness results, and also give all solutions when the number of terms in the sum considered is at most ten.
Primitive prime factors in second-order linear recurrence sequences
Problemi diofantei.
Products of integers in short intervals
Random Thue and Fermat equations
We consider Thue equations of the form , and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations of degree at least six.