Doorn 2026 consecutive integers free certain prime factors

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theorem_1_1: The van Doorn–Tang lower bound n_k > exp(log² k / (20 log log k)) for the least n above 2k whose k preceding integers avoid every prime in (k, 2k); the first superpolynomial bound for Problem 451, from an arXiv preprint whose declaration credits the idea to an AI model and the Lean formalization to an automated prover.

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Wouter van Doorn, Quanyu Tang, Consecutive integers free of certain prime factors. arXiv preprint (2026). arXiv:2606.19863.

Write n_k for the least n > 2k such that no prime in (k,2k) divides any of n-k, ..., n-1 (Erdős Problem #451). The authors prove n_k > exp(log^2 k / (20 log log k)) for all sufficiently large k, which settles Erdős's expectation that n_k eventually exceeds every power of k. Erdős and Graham had written that they could prove n_k > k^{1+c}, but the authors know of no proof of a non-trivial bound in the literature (p. 1). The precise statement is Theorem 1.1, which shows that for large k every n with 2k < n below that threshold fails. The method imitates Konyagin's lower-bound proof for the least prime factor of a binomial coefficient. Write K for the number of integers m in the short interval I = (k, k + k^theta) with n/m closer than k^(theta-1) to an integer; each prime of I dividing none of n-k, ..., n-1 is such an m, so K = o(k^theta/log k) suffices (p. 2). Section 3 bounds K directly, and Sections 5 and 6 bound it through a multi-term estimate (Theorem 4.1, a Konyagin-type inequality with free parameters r and lambda); primes in short intervals come from Baker–Harman–Pintz. The argument is split by the size of n into small, medium, medium-large and large ranges (Sections 2, 3, 5, 6). The paper's declaration of AI usage (p. 2) calls the text "completely human-written", credits an AI model (ChatGPT 5.5 Pro) with the idea of applying Konyagin's argument to this problem, points to that model's original write-up in a public repository, and says that an automated prover (Aristotle, from Harmonic) produced Lean formalizations of every theorem, Konyagin's included, self-contained apart from the Baker–Harman–Pintz input. The card records these as the paper's own declarations and claims no independent check of the argument. The paper does not mention Problem 961: its n_k concerns prime factors in the interval (k, 2k), so Theorem 1.1 neither bounds nor is bounded by that problem's f(k), the least block length forcing a prime factor greater than k; the row below records it as adjacent context only.

The copy read for this card is arXiv:2606.19863v1 (18 June 2026; five pages; complete text layer), the only arXiv version on 2026-09-18, with no journal reference on arXiv and no Crossref record; page numbers below are the preprint's. The introduction (p. 1) quotes Erdős's 1979 Acta paper with the footnote "Notation slightly altered to match notation in [7]": the 1979 text writes the block as $m_n+i$, $1\le i\le n$, while the quotation and the 1980 monograph use the $n_k$ of the site's statement. Read status: claims checked for Theorem 1.1 (p. 1, page image and text layer) and Theorem 4.1 (p. 3, text layer); the proof (Sections 2--6, pp. 2--5) was read for its structure and not checked step by step; nothing here is independently reviewed. The statement is on theorem_1_1. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2606.19863), every other right reserved.

Source: https://arxiv.org/abs/2606.19863.

Bears on. #451: Theorem 1.1 is the problem's first superpolynomial lower bound, $n_k>\exp(\log^2k/(20\log\log k))$ for all large $k$, for exactly the problem's $n_k$ (abstract and Theorem 1.1, p. 1); the site's commentary records it. #961: adjacent context only; the prime-factor cutoff $(k,2k)$ differs from that problem's "a prime greater than $k$", so the theorem says nothing about its $f(k)$.

Results to transcribe.

• Theorem 1.1 (p. 1): fix theta in (2/5, 3/5) such that (k, k + k^theta) contains

  _theta k^theta/log k primes for all large k (Baker–Harman–Pintz supply such a theta); then for all large k, every n with 2k < n <= exp(log^2 k / (20 log log k)) has a prime of (k, k + 3k^theta) dividing one of n-k, ..., n-1, so n_k exceeds that bound.

• Theorem 4.1: Konyagin-type bound: for 2 <= r <= k^{1-theta}/2 and lambda >= 1, K << k^theta((n r! lambda^r/k^{r+1})^{1/(2r-1)} + (k^{r+theta}/(n r! lambda^r))^{1/(r-1)} + ((r+1)lambda/k)^{1/(2r)}) + r lambda.
• Upper bound remark: n_k <= product of primes in (k,2k) = exp((1+o(1))k) by the prime number theorem; heuristics suggest the truth is exp(Theta(k/log k)).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.