Plus grand facteur premier de valeurs de polynômes aux entiers

R. de la Bretèche¹

¹ Institut de Mathématiques de Jussieu UMR 7586 Université Paris–Diderot UFR de Mathématiques, case 7012 Bâtiment Sophie Germain 75205 Paris Cedex 13, France

Acta Arithmetica, Tome 169 (2015) no. 3, pp. 221-250

Résumé

Let $P^+(n)$ denote the largest prime factor of the integer $n$. Using the Heath-Brown and Dartyge methods, we prove that for any even unitary irreducible quartic polynomial $\varPhi $ with integral coefficients and the associated Galois group isomorphic to $V_4$, there exists a positive constant $c_\varPhi $ such that the set of integers $n\leq X$ satisfying $P^+ ( \varPhi (n) )\geq X^{1+c_\varPhi } $ has a positive density. Such a result was recently proved by Dartyge for $\varPhi (n)=n^4-n^2+1$. There is an appendix written with Jean-François Mestre.

Détail

DOI : 10.4064/aa169-3-2

Mots-clés : denote largest prime factor integer using heath brown dartyge methods prove even unitary irreducible quartic polynomial varphi integral coefficients associated galois group isomorphic there exists positive constant varphi set integers leq satisfying varphi geq varphi has positive density result recently proved dartyge varphi n there appendix written jean fran ois mestre