---

Claim. Theorem 1.1 of Cécile Dartyge and James Maynard, On the largest prime factor of quartic polynomial values: the cyclic and dihedral cases, states: for a monic irreducible quartic $P\in\mathbb Z[X]$ whose Galois group is $C_4$ or $D_4$ there is a constant $c_P>0$ such that, for $x>x_0(P)$,

$$\#\{x<m\leq2x:\ P^+(P(m))\geq x^{1+c_P}\}\gg x.$$

Taking $x=n/2$ gives an $m\le n$ whose value $P(m)$ has a prime factor at least $(n/2)^{1+c_P}$, so $F_P(n)\gg_P n^{1+c_P}$ in the notation of Problem 976. The statement and this consequence are recorded on the theorem card.

Covers. The first question for monic irreducible quartics with Galois group $C_4$ or $D_4$. It covers no other quartic and does not give $F_P(n)\gg n^4$.

Depends on. No page of this wiki.

Acceptance. Refereed: the Journal of the European Mathematical Society accepted the paper on 12 October 2023 and published it online on 31 January 2025 (DOI 10.4171/JEMS/1586). The site labels the problem OPEN.