(Universite Paris Diderot, Institut de Mathematiques de Jussieu-Paris Rive Gauche)

*Largest prime factor of integer value of polynomial of degree 4*

Abstract

Let $P^+(n)$ denote the largest prime factor of the integer n. Using
Heath-Brown and Dartyge methods, we prove that for all even unitary
irreducible quartic polynomials ^H $F$ with integral coefficients and
an associated Galois group isomorphic to $V_4$, there exists a
positive constant c >0 such that the set of integers $n\leq x$
satisfying $$P^+( F(n) )\leq x^{1+c_F}$$
has a positive density. Such a result was recently proved by Dartyge
for $F(n)=n^4-n^2+1$.