Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

Streszczenie

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.