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 Φ with integral coefficients and the associated Galois group isomorphic to V₄, there exists a positive constant such that the set of integers n ≤ X satisfying has a positive density. Such a result was recently proved by Dartyge for Φ(n) = n⁴ - n² + 1. There is an appendix written with Jean-François Mestre.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-3-2,
author = {R. de la Bret\`eche},
title = {Plus grand facteur premier de valeurs de polyn\^omes aux entiers},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {221-250},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-3-2}
}
R. de la Bretèche. Plus grand facteur premier de valeurs de polynômes aux entiers. Acta Arithmetica, Tome 168 (2015) pp. 221-250. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-3-2/