Let $f\in \mathbb{Z}\left[x\right]$ be a polynomial of degree $d\ge 3$ without roots of multiplicity $d$ or $(d-1)$. Erdős conjectured that, if $f$ satisfies the necessary local conditions, then $f\left(p\right)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy.The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations.

We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A₅ + A₁.

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 ?

We consider Diophantine quintuples a, b, c, d, e. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most $5.441·{10}^{26}$ Diophantine quintuples.

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field $k$ and any finite set of places $S$ of $k$, one can effectively compute the set of isomorphism classes of hyperelliptic curves over $k$ with good reduction outside $S$. We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus $g$ would imply an effective version of Siegel’s theorem for integral points on...

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....

We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.

This paper is concerned with the density of rational points on the graph of a non-algebraic pfaffian function.

We consider an irreducible curve $𝒞$ in $E^n$, where $E$ is an elliptic curve and $𝒞$ and $E$ are both defined over $\overline{\mathbb{Q}}$. Assuming that $𝒞$ is not contained in any translate of a proper algebraic subgroup of $E^n$, we show that the points of the union $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, where $A$ ranges over all proper algebraic subgroups of $E^n$, form a set of bounded canonical height. Furthermore, if $E$ has Complex Multiplication then the set $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, for $A$ ranging over all algebraic subgroups of $E^n$ of codimension at least $2$, is finite. If $E$ has no Complex Multiplication...