---

Fix $f\in\mathbb Z[X]\setminus\{0\}$ and $\varepsilon_0\in(0,1/100)$. Let $C_0$ be the constant from Lemma 2.2 of the paper, which depends only on $f$: for every prime $p$ and every interval $J\subset[1,p)$ with $|J|\geq1$, at most $C_0|J|^{2/3}$ integers $n\in J$ satisfy $p\mid n!+f(n)$. The paper's $O$-constants may depend on $f$ and $\varepsilon_0$, but not on $x$.

Statement

Suppose that $x$ is sufficiently large in terms of $f$ and $\varepsilon_0$. Let $p$ be prime, and let $t$ be an integer satisfying

$$\left(\frac{10}{\varepsilon_0}\right)^{100} \leq t\leq C_0x^{2/3}.$$

Let $J$ be an interval with

$$J\subset[\varepsilon_0x,\min\{x,p\}),$$

and let $n_1,\ldots,n_t$ be distinct integers in $J$. Then

$$(\log p) \sum_{t'=\left\lceil\frac{1+\varepsilon_0}{2}t\right\rceil}^{t} \min_{1\leq j\leq t'} \operatorname{ord}_p(n_j!+f(n_j)) \leq \frac{|J|}{2}\log t+ \frac{x\log x}{t^{0.98}}+O(x). \tag{2.14}$$

Source and proof pointer

This is Lemma 2.7 and equation (2.14) on physical p. 6 of the selected arXiv:2103.14894v1 PDF. Its proof occupies physical pp. 7--8 and ends immediately before Section 3.

The source proof uses Definition 2.3, Lemma 2.4, Heath-Brown's prime-gap bound as Lemma 2.5, and Corollary 2.6. Those steps are not transcribed here. This page records the exact statement and dependency pointer only, with no complete-proof or independent-verification claim.

Bears on

No Erdős problem page directly. It is the new input to the proof of Theorem 1.1, whose page states that theorem's relation to Problem 977.