Problem 451

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

Number theory

Wiki page

Status

Open

No claim settles this problem.

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 (2026), 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.

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 and Tang 2026; 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.
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: Erdős and Graham 1980.
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).

Problem 451

Status

Proof claims

Discussion