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For an integer $m>1$, let $P(m)$ denote its greatest prime divisor.

Statement

For every polynomial $f\in\mathbb Z[X]\setminus\{0\}$,

$$\limsup_{n\to\infty}\frac{P(n!+f(n))}{n} \geq1+9\log2\approx7.238. \tag{1}$$

Moreover, fix $\varepsilon>0$ and call a positive integer $n$ good when

$$n!+f(n)>1 \quad\text{and}\quad P(n!+f(n))>(1+9\log2-\varepsilon)n.$$

The good integers then have positive lower asymptotic density: for some $\delta>0$, at least $\delta N$ of the integers $n\leq N$ are good once $N$ is large.

Source and proof pointer

Theorem 1.1 begins on physical p. 1 and its positive-density clause continues at the top of physical p. 2 of the selected arXiv:2103.14894v1 PDF. Its proof is Section 3, physical pp. 8--11.

The proof relies on the paper's preliminary setup and, in particular, the new Lemma 2.7. The theorem proof and the preliminary lemmas are not transcribed here. This page records a statement and proof pointer only and carries no complete-proof or proof-verification claim.

Bears on

• Problem 977 (context only): the case $f=1$ gives $\limsup_{n\to\infty}P(n!+1)/n\geq1+9\log2$ for the factorial sequence that the problem's catalog remarks mention. It is a finite limsup lower bound, so it does not show that $P(n!+1)/n\to\infty$, and it says nothing about $P(2^n-1)/n$, the quantity the problem asks about.