We consider Thue equations of the form $a{x}^k+b{y}^k=1$, 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 $a{x}^k+b{y}^k+c{z}^k=0$ of degree at least six.

We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find...

Dans des travaux profonds, W. Ljunggren a montré que, pour $a\>0$ donné, les équations diophantiennes $x^4-a{y}^2=1$ and $x^2-a{y}^4=1$ ont au plus $1$ ou $2$ solutions non triviales. Par des méthodes élémentaires, je réponds ici à la question : pour quelles valeurs de $a$, premières ou analogues, ont-elles des solutions non-triviales ?