Partitions in the prime number maze
In this article we compute the th power values of the quadratic polynomials with negative squarefree discriminant such that is coprime to the class number of the splitting field of over . The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer...
Récemment, B. Green et T. Tao ont montré que : l’ensemble des nombres premiers contient des progressions arithmétiques de toutes longueurs répondant ainsi à une question ancienne à la formulation particulièrement simple. La démonstration n’utilise aucune des méthodes “transcendantes” ni aucun des grands théorèmes de la théorie analytique des nombres. Elle est écrite dans un esprit proche de celui de la théorie ergodique, en particulier de celui de la preuve par Furstenberg du théorème de Szemerédi,...
In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that for sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor on its right hand side by the factor for a given , and by replacing the numerical factor by a given positive . Finally, we introduce and study inequalities analogous...