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Claim. With $q(n,k)$ the least prime not dividing $\prod_{1\le i\le k}(n+i)$, there are infinitely many $n$ with $q(n,\log n)\gg\log n,\log\log n/\log\log\log n$. Since the right side exceeds $(2+\epsilon)\log n$ for every fixed $\epsilon$ once $n$ is large, this answers the question of Problem 457 yes, for every constant in place of $2$, in a form sharper than the construction on the sibling page Barreto. The write-up's Theorem 1 uses $f\gg g$ to mean $f\ge cg$ for an absolute constant $c>0$; its proof (p. 5) gives $q(n,\lfloor\log n\rfloor)>(\frac12+o(1))\log n\log\log n/\log\log\log n$, the bound the site's commentary credits to Tao's sketch.

Submission note. Posted to the site's forum by Nat Sothanaphan on 3 March 2026:

I have GPT with near-autonomous process expand this into a writeup. We do get $q(n, \log n) \gg \log n \log \log n/\log \log \log n$ for infinitely many $n$. However, GPT is unable to derive the $(1-o(1)) \log n \log \log n/\log \log \log n$ version. Of course, this may be doable, just that GPT does not know how.

Method. The write-up expands the elaboration Tao sketched in the thread on 2 March 2026: the primes up to $k=\log n$ divide the block automatically, and for the primes in $(k,Ak]$ it suffices that $n$ lie within $k/2$ of a multiple of each, which Dirichlet's approximation theorem achieves with $n$ small enough to allow $A$ of order $\log k/\log\log k$ while $n\le e^k$. The poster reports that GPT could not reach the constant $(1-o(1))$ in front of $\log n\log\log n/\log\log\log n$; Tao's reply of the same day notes the factor $1/2$ lost in passing from a symmetric block to $\prod_{i=1}^k(n+i)$, so that the method's natural limit is $q(n,\log n)\ge(\frac12-o(1))\log n\log\log n/\log\log\log n$, as GPT worked out. The write-up works with the symmetric block and a weighted Dirichlet box lemma that places $m$ in $[B,B^2]$; its Remark 1 attributes the lost constant to that range and says that a solution in $[B,B^{1+o(1)}]$ would plausibly give $1-o(1)$.

Claimant. The forum account Nat Sothanaphan, who posted the write-up and states that GPT produced it in what the claimant calls a near-autonomous process; the write-up's disclaimer names the system as GPT-5.2 Thinking.

Standing. Claimed. The site's label PROVED (LEAN) and its commentary credit GPT-5.2 Pro's construction, prompted by Barreto, and record Tao's elaboration as a sketch in the comments; neither names this write-up, and no paper, refereed publication or Lean file of it exists. Tao's thread reply thanks the poster for fleshing out the details and confirms the method's limit, which is endorsement in a thread, not a review record. Nothing was reviewed here.

Depends on. No page of this wiki.