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Plus grand facteur premier de valeurs de polynômes aux entiers

Volume 169 / 2015

R. de la Bretèche Acta Arithmetica 169 (2015), 221-250 MSC: Primary 11N32. DOI: 10.4064/aa169-3-2

Abstract

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.

Authors

  • R. de la BretècheInstitut de Mathématiques de Jussieu
    UMR 7586
    Université Paris–Diderot
    UFR de Mathématiques, case 7012
    Bâtiment Sophie Germain
    75205 Paris Cedex 13, France
    e-mail

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