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Source. Theorem 1, p. 1, of P. Chojecki, A note on Erdős Problem #1201, preprint note dated 30 April 2026, as identified on the source card. The print carries no byline.

Statement

Notation (p. 1). For $m\ge2$, $P^+(m)$ is the largest prime divisor of $m$, and $P^+(1)=1$. For $A\subseteq\mathbb N$, $\overline d(A)$ and $\underline d(A)$ are the $\limsup$ and the $\liminf$ as $N\to\infty$ of $|A\cap[1,N]|/N$. The note sets

$$\mathcal G_{\varepsilon,k}=\{n\in\mathbb N:P^+(n(n+1)\cdots(n+k))>n^{1-\varepsilon}\}.$$

Theorem 1 (p. 1). For every $\varepsilon>0$,

$$\lim_{h\to\infty}\overline d\Bigl\{n\in\mathbb N: P^+\Bigl(\prod_{j=0}^{h-1}(n+j)\Bigr)\le n^{1-\varepsilon}\Bigr\}=0 .$$

Consequently, for every $\varepsilon,\eta>0$ there is a $k$ with $\underline d(\mathcal G_{\varepsilon,k})\ge1-\eta$.

The theorem bounds the lower density of $\mathcal G_{\varepsilon,k}$; it does not assert that the natural density of $\mathcal G_{\varepsilon,k}$ exists. The quantitative form proved (p. 4) is $\overline d(\mathcal B_{\varepsilon,h})\le C(\log h)^{1/3}/(\delta^2h^{\delta/25})$ for $0<\varepsilon<1$, where $\mathcal B_{\varepsilon,h}$ is the set inside the limit above, $C$ is the absolute constant of Theorem 2, $\delta=(1-\rho(1/\beta))/4$ with $\beta=1-\varepsilon/2$ and $\rho$ the Dickman--de Bruijn function, and $h$ is large enough that $C_0\log\log h/\log h\le\delta$ (inequality (2), p. 2).

Proof pointer

Pp. 2--4. The case $\varepsilon\ge1$ is trivial. For $0<\varepsilon<1$ the note applies Theorem 2 (p. 2), its half-open form of Theorem 1 of Matomäki and Radziwiłł (Multiplicative functions in short intervals, Ann. of Math. (2) 183 (2016), 1015--1056), to the completely multiplicative indicator $f_X$ of the integers whose prime factors are all at most $X^\beta$. By the Dickman--de Bruijn count (1), the mean of $f_X$ over $[X,2X)$ is $\rho(1/\beta)+o(1)$, which is less than $1$. Every $n\in[X,2X]$ in the bad set makes all of $f_X(n),\dots,f_X(n+h-1)$ equal to $1$ once $X$ is large, since $(2X)^{1-\varepsilon}<X^\beta$, so such $n$ lie in the exceptional set of Theorem 2. This gives the dyadic bound (5) (p. 3), and a dyadic decomposition of $[1,N]$ turns it into the upper-density bound (p. 4). The second assertion follows with $h=k+1$ by taking complements.

Dependencies

Theorem 2 (p. 2), an external input quoted from Matomäki and Radziwiłł with absolute constants $C,C_0>0$ uniform in $f$, $h$, $X$ and $\delta$; the Dickman--de Bruijn estimate (1) (p. 2), cited to Tenenbaum's Introduction to Analytic and Probabilistic Number Theory, Chapter III.5. Read depth: claims checked; the statement, notation and the proof's steps were read on the print, and the proof was not checked independently.

Bears on

• Problem 1201: the problem asks whether for every $\epsilon,\eta>0$ some $k$ makes the density of the $n$ with $P(n(n+1)\cdots(n+k))>n^{1-\epsilon}$ at least $1-\eta$. The second assertion of the theorem gives this with lower density in place of density. The note says that it settles the problem as stated on the Erdős Problems website (p. 1), and calls the result an immediate but apparently unrecorded consequence of the Matomäki--Radziwiłł theorem (pp. 1 and 4).