La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant $D$, sous l’action de ${\mathrm{SL}}_2$ par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant $D$. Les domaines fondamentaux associés permettent calculs explicites et évaluation d’ordres moyens. Je présenterai les lois de composition supérieures découvertes par M. Bhargava à partir de la classification des espaces vectoriels préhomogènes réguliers,...

Let K be a finite Galois extension of the field ℚ of rational numbers. We prove an asymptotic formula for the number of Piatetski-Shapiro primes not exceeding a given quantity for which the associated Frobenius class of automorphisms coincides with any given conjugacy class in the Galois group of K/ℚ. In particular, this shows that there are infinitely many Piatetski-Shapiro primes of the form a² + nb² for any given natural number n.

Let $S$ be a linear integer recurrent sequence of order $k\ge 3$, and define $P_S$ as the set of primes that divide at least one term of $S$. We give a heuristic approach to the problem whether $P_S$ has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that $P_S$ has positive lower density for “generic” sequences $S$. Some numerical examples are included.