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A+ CATEGORY SCIENTIFIC UNIT

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 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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