Problem 451

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claims/: The 1 claim page of Problem 451, one per claimant's result; the problem's standing derives from them.

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Statement. Estimate $n_k$, the smallest integer $>2k$ such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.

Formulation. The site's wording on 2026-09-18 (page last edited 21 June 2026). The product runs over the $k$ integers $n_k-k,\ldots,n_k-1$ preceding $n_k$; the condition $n_k>2k$ places that block above $k$, without which $n_k=k+1$ would work trivially (the product $k!$ has no prime factor in $(k,2k)$). The 1980 monograph's sentence (below) omits the condition, and the site's and the 2026 paper's wording supply it. Erdős's 1979 paper asks the same question in a shifted form, for the smallest $m_n\ge n$ such that $\prod_{1\le i\le n}(m_n+i)$ has no prime factor $p$ with $n<p<2n$: both ask for the least start $s>k$ of a block of $k$ consecutive integers with no prime factor in $(k,2k)$, the site's $n_k$ being $s+k$ and the 1979 $m_k$ being $s-1$, so $n_k=m_k+k+1$ (a substitution made here). The site's source keys are [Er79d, p. 78] and [ErGr80, p. 89]; its commentary cites [vDTa26] and points to Problem 1095.

Status. Open. The estimate is not settled: the best lower bound claimed is $n_k>\exp(\log^2k/(20\log\log k))$ for all sufficiently large $k$, Theorem 1.1 of van Doorn and Tang (arXiv:2606.19863v1, 18 June 2026; a preprint, taken into the site's commentary on 21 June 2026), which would confirm Erdős's expectation that $n_k$ grows faster than every power of $k$ and is recorded as a claimed partial claim on the van Doorn--Tang claim page, since the site's commentary on a problem it labels OPEN is no acceptance; the best upper bound is the elementary $n_k\le\prod_{k<p<2k}p=e^{(1+o(1))k}$; Erdős expected $n_k<e^{\epsilon k}$ for every $\epsilon>0$, and a heuristic in the thread and the paper puts the truth at $\exp(\Theta(k/\log k))$. Before 2026 the only lower bound was the monograph's unproved sentence "We can prove $n_k>k^{1+c}$". The preprint's declaration credits the idea of its argument to ChatGPT 5.5 Pro and its Lean formalization to Aristotle, Harmonic's automated prover; the provenance is recorded below without changing the standing of the bound, which is progress on an open question, not its resolution. No source narrowing the gap further was found in the search whose scope the Current assessment records; this is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/451, accessed 2026-09-18: the problem page (OPEN, with the site's note that the problem cannot be settled by a finite computation; last edited 21 June 2026; source keys [Er79d, p. 78], [ErGr80, p. 89]; commentary citing [vDTa26] and Problem 1095; a thanks line naming Adenwalla and van Doorn), its six-comment discussion thread (2 October 2025 to 19 June 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #451, https://www.erdosproblems.com/451, accessed 2026-09-18.

References.

• [vDTa26] van Doorn, W. and Tang, Q., Consecutive integers free of certain prime factors. arXiv:2606.19863v1 (18 June 2026), 5 pp.; Theorem 1.1 and the upper bound, p. 1; the declaration of AI usage, p. 2; Theorem 4.1, p. 3. Library home: doorn_2026_consecutive_integers_free_certain_prime_factors; result page Theorem 1.1.
• [Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. 33 (1979), no. 1--2, 71--80, DOI 10.1007/BF01903382 (Crossref record accessed); Section 3, printed p. 78. Library home: erdos_1979_unconventional_problems_number_theory.
• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980); printed p. 89. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Ko99] Konyagin, S. V., Estimates of the least prime factor of a binomial coefficient. Mathematika 46 (1999), 41--55. Not held; the method source of [vDTa26] (its Theorem 2 is applied in Theorem 4.1) and the best lower bound for Problem 1095, as the paper states.
• [BHP01] Baker, R. C., Harman, G. and Pintz, J., The difference between consecutive primes, II. Proc. London Math. Soc. (3) 83 (2001), no. 3, 532--562. Not held; the paper's source for primes in the short interval $(k,k+k^\theta)$.
• [OEIS] Kalogeropoulos, G., Sequence A386620, The On-Line Encyclopedia of Integer Sequences (2025; entry last modified 22 August 2026, server time): $n_k$ for $k\le209$ (b-file by W. A. Carney, with a C++ program).

Formalization. None in formal-conjectures: no file ErdosProblems/451.lean exists in google-deepmind/formal-conjectures at the head of main on 2026-09-18 (the directory FormalConjectures/ErdosProblems/ had 673 entries, none for this problem, and the recursive tree none either), and the problem page's indicator shows no formalized statement. The community database (teorth/erdosproblems) records the problem open (last changed 31 August 2025), the statement not formalized, formal_status unformalized, OEIS A386620 and no formal-proof URL. The external Lean development of the 2026 bound is described under "External artifacts" below; nothing was built.

Current assessment

The question (site formulation of 2026-09-18). The statement above; OPEN and, by the site's note, not settled by any finite computation, last edited 21 June 2026. The commentary, in this page's words: it quotes the monograph's sentence that $n_k>k^{1+c}$ is provable and that much more is surely true; it reports Erdős's 1979 expectation that $n_k$ stays below $e^{o(k)}$ yet exceeds every fixed power $k^d$; it credits Adenwalla with the easy upper bound $n_k\le\prod_{k<p<2k}p=e^{O(k)}$; and it records that van Doorn and Tang [vDTa26], by an argument it attributes to GPT 5.5 Pro and to Tang, proved $n_k>\exp(c(\log k)^2/\log\log k)$ for some $c>0$, pointing to Problem 1095. The thread, oldest first: 2 October 2025 (the account Quanyu Tang), a heuristic: the density of admissible $n$ is $D_k=\prod_{k<p<2k}(1-k/p)$ with $\log(1/D_k)=(\log4+o(1))k/\log k$, and $\log n_k\le(\log4)k/\log k$ for $k\le50$, with a note in a GitHub repository; 2 November 2025 (the account Giorgos Kalogeropoulos), the OEIS entry A386620 is published (the site was updated); 26 April 2026 (Quanyu Tang), a note produced by GPT 5.5 Pro after many prompting rounds claiming $n_k>\exp(c(\log k)^2/\log\log k)$, which the poster said had passed several AI-based checks, posted with a request for independent checks; 27 April 2026 (the account Nat Sothanaphan), a routine model check, reported in a shared transcript not cited here, found no issue; 19 June 2026 (the account Woett, whom the site names as Wouter van Doorn), the arXiv paper, presented as the authors' human-written version of the AI-generated note, with the parameter $\theta=21/40$, the bound (2) $n_k>\exp(\log^2k/(20\log\log k))$, a Lean formalization self-contained apart from the Baker--Harman--Pintz input, the connection to Konyagin's theorem and Problem 1095, and credit to the model for the bound (the site was updated); 19 June 2026 (Nat Sothanaphan), congratulations. The proof-claim tab is empty.

The origins. [ErGr80], printed p. 89: "Let $n_k$ denote the smallest integer for which $\prod_{i=1}^k(n_k-i)$ has no prime factor in $(k,2k)$. We can prove $n_k>k^{1+c}$ but no doubt much more is true." No proof or reference accompanies the sentence, and the condition $n_k>2k$ is not printed (Formulation above). [Er79d], printed p. 78, in Section 3: "Several times during my long life, I was led to questions of the following type. Estimate, as well as you can, the size of the smallest integer $m_n\ge n$ for which $\prod_{1\le i\le n}(m_n+i)$ has no prime factor $p$ satisfying $n<p<2n$. I would expect that $m_n>n^k$ for every $k$ if $n>n_0(k)$, but that $m_n<e^{\varepsilon n}$ for every $\varepsilon>0$ if $n>n_1(\varepsilon)$. However, I could prove nothing non-trivial." The 2026 paper quotes the passage in the site's notation, with the footnote "Notation slightly altered to match notation in [7]" (the monograph).

The 2026 lower bound. Theorem 1.1 of [vDTa26] (p. 1): with $\theta\in(2/5,3/5)$ a constant such that $(k,k+k^\theta)$ contains $\gg_\theta k^\theta/\log k$ primes for all large $k$ (by [BHP01]), for all sufficiently large $k$ and all $n$ with $2k<n\le e^{\log^2k/(20\log\log k)}$ the product $(n-k)\cdots(n-1)$ is divisible by some prime $p\in(k,k+3k^\theta)$. Since $3k^\theta<k$ for large $k$, this is $n_k>e^{\log^2k/(20\log\log k)}$, the abstract's statement, and it would settle Erdős's expectation $n_k>k^d$ for every $d$. The proof (pp. 2--5) imitates Konyagin's argument for the least prime factor of a binomial coefficient: if no prime $p$ in $(k,k+k^\theta)$ divides the product then $n/p$ is within $k^{\theta-1}$ of an integer, and, after settling the smallest of four ranges of $n$ by exhibiting the factor directly, the paper bounds the number of $m$ in that interval with $\|n/m\|<k^{\theta-1}$ in the other three, the last two through a Konyagin-type inequality (Theorem 4.1, p. 3). Read depth: claims checked for Theorem 1.1 and Theorem 4.1; the proof was read for its structure and not checked step by step; no step is independently reviewed here. Acceptance evidence: the site's commentary (21 June 2026), and the bound carries the preprint qualification: an author-recorded theorem taken into the site's commentary, without refereed publication or independent review found; the site labels the problem OPEN, so the commentary is no acceptance. It is recorded as a claimed partial claim on the van Doorn--Tang claim page. Provenance, recorded not judged: the paper's declaration of AI usage (p. 2) says the text is "completely human-written", that the core idea of applying Konyagin's argument was conceived by ChatGPT 5.5 Pro, whose original write-up the authors keep in a public repository "for transparency's sake", and that Lean formalizations of all theorems, self-contained apart from [BHP01], were produced by Aristotle, Harmonic's automated theorem prover; the site's commentary attributes the argument to GPT 5.5 Pro and to Tang.

The upper bound and the expected size. If $n\equiv0$ modulo every prime $p\in(k,2k)$ then $n-i\equiv-i\not\equiv0\pmod p$ for $1\le i\le k$, so $n=\prod_{k<p<2k}p$ is admissible once it exceeds $2k$, giving $n_k\le\prod_{k<p<2k}p=e^{(1+o(1))k}$ by the prime number theorem (the site's easy upper bound, credited to Adenwalla; stated on p. 1 of the paper; the one-line verification is made here). Erdős expected $n_k<e^{\varepsilon k}$ for every $\varepsilon>0$, which is open. The heuristic of the thread's first comment and of the paper's introduction, $n_k\asymp D_k^{-1}$ with $\log(1/D_k)=(\log4+o(1))k/\log k$, suggests $\log n_k=\Theta(k/\log k)$; the OEIS entry A386620 (accessed 2026-09-18; not recomputed here) lists $n_k=3,6,9,20,13,21,21,22,65,220,51,338,133,321,\ldots$ for $k=1,2,\ldots$, non-monotone in $k$, with a b-file to $k=209$.

The bounds map. $\exp(\log^2k/(20\log\log k))<n_k\le e^{(1+o(1))k}$ for all large $k$, the lower bound a 2026 preprint and the upper bound elementary; expected $\log n_k\asymp k/\log k$. Erdős's shifted $m_k$ obeys $m_k=n_k-k-1$, so the same bounds hold for it. Problem 1095, the least $n>k+1$ such that $\binom nk$ has no prime factor at most $k$, has the analogous bounds $k^{1+c}<n<e^{(1+o(1))k}$ (Ecklund, Erdős and Selfridge) with Konyagin's $e^{c\log^2k}$ below, as the paper's introduction states; the two problems are linked by the method, not by an implication.

External artifacts (not built). The thread links ErdosProblem451.lean in the repository Woett/Lean-files, at the repository's head of 2026-09-10 (the file last changed on 2026-06-19, the commit the claim page's link pins; 330,548 bytes, 4,693 lines; accessed 2026-09-18; Lean v4.28.0). Its header says that it proves the best known lower bound for Problem 451 and names the prover that produced it and the model that found the argument, Aristotle and GPT 5.5 Pro as the paper's declaration names them. It declares one axiom, bhp, the Baker--Harman--Pintz count $C\,k^\theta/\log k$ of primes in $(k,k+k^\theta)$ for large $k$ with $\theta=21/40$, and proves theorem main_theorem : ∃ k₀ : ℕ, ∀ k : ℕ, k₀ ≤ k → ∀ n : ℤ, 2 * (k : ℤ) < n → (n : ℝ) ≤ Real.exp ((Real.log k) ^ 2 / (20 * Real.log (Real.log k))) → ∃ p : ℕ, p.Prime ∧ (k : ℝ) < p ∧ (p : ℝ) < (k : ℝ) + 3 * (k : ℝ) ^ theta ∧ (p : ℤ) ∣ Pprod k n, Theorem 1.1 with the paper's four cases, and Konyagin's theorem along the way as konyagin_thm; it contains no sorry and ends with #print axioms main_theorem, whose output is not recorded. Nothing was built or kernel-checked here, and the theorem holds relative to the declared axiom. The repositories QuanyuTang/Notes-on-Erdos-Problem-451 (head of 2026-04-26: the heuristics note of October 2025 and the April 2026 lower-bound note, whose TeX source states the theorem $n_k>\exp(c(\log k)^2/\log\log k)$ for an absolute $c>0$ and all large $k$) and the model's original write-up named in the paper's reference [10] are leads; of the April note, only the theorem statement from its TeX source is recorded here.

Search scope. None of the routes below found a journal version of [vDTa26], an independent review, a dispute, or a bound narrowing the gap.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures directory listing and tree at the head of main on 2026-09-18 (no file); the community database that day.
• arXiv: the abstract page and API record of 2606.19863 (v1 only, 18 June 2026; a comments field giving five pages; no journal reference); the API queries abs:"consecutive integers" AND abs:"prime factor" AND (abs:Erdos OR abs:Erdős) (two records, neither on $n_k$) and all:"Erdős Problem" AND (all:451 OR all:457 OR all:961 OR all:962 OR all:1181) (no records).
• Crossref: a bibliographic query for the paper's title (no record; three unrelated papers on prime factors of consecutive integers); the record of [Er79d].
• Semantic Scholar: the citation list of arXiv:2606.19863 (empty); the paper endpoint answered HTTP 429 and was not retried.
• GitHub API: Woett/Lean-files (head, the file, its history) and QuanyuTang/Notes-on-Erdos-Problem-451 (head, listing, the TeX source of the April note).
• OEIS: the JSON record of A386620.
• The primary sources: [vDTa26] pp. 1--5; [Er79d] p. 78; [ErGr80] p. 89.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not held: [Ko99], [BHP01]. Ecklund--Erdős--Selfridge 1974 (the paper's [5]: A new function associated with the prime factors of $\binom nk$, Math. Comp. 28 (1974), 647--649) is not held either; its card is ecklund_1974_new_function_associated_prime_factors (the abstract's bounds $k^{1+c}<g(k)<\exp(k(1+o(1)))$ on printed p. 647, and the lower bound (6) with its proof from the Erdős--Selfridge prime-factor theorem on printed p. 648, on the result page Inequality (6); the card's Bears-on row names Problem 1095 only).

Remaining gaps. (1) The lower bound is a preprint result whose argument is credited to an AI model, with a Lean development not built; a refereed version or an independent review is the reopening condition for its qualification. (2) The gap between $\exp(\log^2k/(20\log\log k))$ and $e^{(1+o(1))k}$ is the problem; Erdős's $e^{\varepsilon k}$ expectation and the $k/\log k$ heuristic are unproved. (3) The monograph's "We can prove $n_k>k^{1+c}$" is accompanied by no proof or reference. (4) Proof coverage is at statement level; [Ko99] and [BHP01] are not held.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1979_unconventional_problems_number_theory
• erdos_1979_unconventional_problems_number_theory / section_3
• doorn_2026_consecutive_integers_free_certain_prime_factors
• doorn_2026_consecutive_integers_free_certain_prime_factors / theorem_1_1
• erdos_1980_old_new_problems_results_combinatorial_number_theory