Dans cet article, nous étudions certains invariants liés à la réduction de Hermite-Korkine-Zolotareff des réseaux euclidens (ou des formes quadratiques définies positives).

Let ${𝔖}_n$ denote the symmetric group with $n$ letters, and $g\left(n\right)$ the maximal order of an element of ${𝔖}_n$. If the standard factorization of $M$ into primes is $M=q_1^{{\alpha }_1}{q}_2^{{\alpha }_2}...q_k^{{\alpha }_k}$, we define $\ell \left(M\right)$ to be $q_1^{{\alpha }_1}+q_2^{{\alpha }_2}+...+q_k^{{\alpha }_k}$; one century ago, E. Landau proved that $g\left(n\right)={max}_{\ell \left(M\right)\le n}M$ and that, when $n$ goes to infinity, $logg\left(n\right)\sim \sqrt{nlog\left(n\right)}$.There exists a basic algorithm to compute $g\left(n\right)$ for $1\le n\le N$; its running time is $𝒪\left(N^{3/2}/\sqrt{logN}\right)$ and the needed memory is $𝒪\left(N\right)$; it allows computing $g\left(n\right)$ up to, say, one million. We describe an algorithm to calculate $g\left(n\right)$ for $n$ up to ${10}^{15}$. The main idea is to use the so-called $\ell$-superchampion...

At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :(1)Are there infinitely many primes of the form $n^2+1$?(2)The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.(3)The Twin Prime Conjecture.(4)Does there exist always at least one prime between neighbouring squares?All these...

Dans un corps fini, toute série formelle algébrique en une indéterminée est la diagonale d'une fraction rationnelle en deux indéterminées (Furstenberg 67). Dans cet article, nous donnons une nouvelle preuve de ce résultat, par des méthodes purement combinatoires.

For $E_b:y²=x³+b$, we establish Lang’s conjecture on a lower bound for the canonical height of nontorsion points along with upper and lower bounds for the difference between the canonical and logarithmic heights. These results are either best possible or within a small constant of the best possible lower bounds.

We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of bounded height....

This paper consists of three parts. In the first part we prove a general theorem on the image of a language $K$ under a substitution, in the second we apply this to the special case when $K$ is the language of balanced words and in the third part we deal with recurrent Z-words of minimal block growth.

Soit $K$ un corps complet pour une valuation discrète, de corps résiduel $k$. Lorsque $k$ est fini, la structure de $K_2\left(K\right)$ a été déterminée par C.C. Moore, J.E. Carroll et A.S. Merkurjev. On généralise ici leurs résultats au cas où $k$ est parfait de caractéristique positive $p$. Les résultats principaux sont : $p^n{K}_2\left(K\right)$ est $p$-divisible pour $n$ assez grand (explicite); le groupe $K_2^{\mathrm{top}}\left(K\right)$ de Milnor est discret, explicitement déterminé ; $K_2\left(K\right)$ n’a pas de torsion première à $p$, et sa $p$-torsion est explicitement déterminée. On obtient...