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Claim. The note "The large-prime-divisor route to Erdős Problem #1201" (1 May 2026) works in the range $1/2<\alpha<1$, where an integer of size about $X$ has at most one prime factor above $X^\alpha$. For $0\le j<h$ let $L_{j,X}(n)$ be the number of primes $p>X^\alpha$ dividing $n+j$, and for $J\subseteq\{0,\dots,h-1\}$ let $T_J(X)$ be the sum over $X<n\le2X$ of $\prod_{j\in J}L_{j,X}(n)$. The hypothesis the note names $\mathrm{LPD}(\alpha,h)$ is that $T_J(X)=\lambda_\alpha^{|J|}X+o(X)$ for every such $J$, with $\lambda_\alpha=\log(1/\alpha)$. Its Theorem 1.3: assuming $\mathrm{LPD}(\alpha,h)$, the number of $n\le N$ with $P^+(n(n+1)\cdots(n+h-1))\le n^\alpha$ is $\rho(1/\alpha)^hN+o(N)$, so the natural density of that set exists and equals $\rho(1/\alpha)^h$. With $\alpha=1-\varepsilon$ this gives the set of Problem 1201 the natural density $1-\rho(1/(1-\varepsilon))^h$, which exceeds $1-\eta$ for $h$ large; the natural-density variant recorded on the problem page would follow for $0<\varepsilon<1/2$. Theorem 1.4 says that the two-point case $J=\{i,j\}$ of the hypothesis is equivalent to the asymptotic $\rho(1/\alpha)^2X+o(X)$ for the number of $X<n\le2X$ with both $P^+(n+i)$ and $P^+(n+j)$ at most $X^\alpha$, at every scale $X$. The note's abstract says that it does not claim an unconditional proof of the hypothesis and that the hypothesis is exactly the missing fixed-shift, all-scales input. The statements above are those of the note's Section 1; its proofs have not been checked.

Submission note. Posted to the site's forum by Przemysław Chojecki on 1 May 2026:

Thank you for pointing it out - it seems that proving the existence of the natural density in this case is quite hard. I couldn't really complete it in general though I've got

$$\#\{n\le N:P ^{+}((n)\cdots(n+h-1))\le n^\alpha\} > \sim \rho(1/\alpha)^h N .$$

for $1/2 < \alpha < 1$. This is related to your

work with Teräväinen, work of Wang and a couple of other papers. Here's the note by GPT-5.5 Pro with these results and explicit conjecture that's missing to prove the natural density exists.

Hypothesis. $\mathrm{LPD}(1-\varepsilon,h)$ for every $h$ and every $0<\varepsilon<1/2$: an unproven asymptotic for the joint distribution of large prime divisors of $h$ consecutive integers at every scale. The hypothesis at every $h$ is needed because Theorem 1.3 gives the density $1-\rho(1/(1-\varepsilon))^h$ only at an $h$ where $\mathrm{LPD}(1-\varepsilon,h)$ holds, and reaching $1-\eta$ for every $\eta$ needs an $h$ with $\rho(1/(1-\varepsilon))^h<\eta$, so arbitrarily large $h$; as $\mathrm{LPD}(\alpha,h)$ implies $\mathrm{LPD}(\alpha,h')$ for $h'<h$, the sets $J$ shrinking, that is the hypothesis at every $h$. The note's Section 1 phrases its deduction as using the hypothesis for some $h$ chosen large, which as a statement of the hypothesis is too weak. Even granting the hypothesis for every $h$ and every $\varepsilon<1/2$, the natural-density form of the question for $\varepsilon\ge1/2$ is not reached: the method needs $\alpha>1/2$, and a superset of a set of natural density at least $1-\eta$ need not have a natural density, so the conditional result is also partial in $\varepsilon$. Terence Tao wrote in the thread on 1 May 2026 that the note's results are conditional on an unproven and difficult hypothesis, the one the note names LPD, and the claimant confirmed the conditional nature of the results the same day. The unconditional lower-density statement, the precise Statement, is Chojecki's claim and is not conditional.

Standing. Przemek Chojecki posted the note in the site's thread on 1 May 2026, presenting it as written by GPT-5.5 Pro; Chojecki is the claimant as its submitter, and GPT-5.5 Pro is the system Chojecki names. The note itself carries no byline and, as of 2026-10-07, has no refereed publication, no formalization and no outside review, and the site's label is unchanged (OPEN). The claim is rejected: under the unproven hypothesis LPD it gives the natural-density variant only for $0<\varepsilon<1/2$, which implies the precise Statement only in that range, a result the unconditional claim already covers.

Depends on. Nothing on the wiki; the note's unconditional part rests on the same Matomäki--Radziwiłł input as the claim linked above.