Problem 929

Let $k\geq 2$ be large and let $S(k)$ be the minimal $x$ such that there is a positive density set of $n$ where

n+1,n+2,\ldots,n+k

are all divisible by primes $\leq x$ .

Estimate $S(k)$ - in particular, is it true that $S(k)\geq k^{1-o(1)}$ ?

Source: erdosproblems.com/929

Number theory

Wiki page

Status

Open

No claim settles this problem.

The site labels the problem OPEN. No source approaching $S(k)\ge k^{1-o(1)}$ was found in the search whose scope the Current assessment records, and the site's proof-claim tab was empty on 2026-10-07. One partial claim is recorded: the refereed upper bound of [FGKMT18], inverted below. Lower bounds: the site's Rosser bound $S(k)>k^{1/2-o(1)}$ is Erdős's report in [Er76d] p. 26 ("Rosser proved [13] that $\ell>k^{1/2-\varepsilon}$ ", cited to the Halberstam--Richert book); the stronger $S(k)\gg k^{1/2}$ is Iwaniec's $Y(x)\ll x^2$ read through the equivalence, which is how [Er79d] p. 79 states it ("Iwaniec's result $B(n)>c\sqrt n$ is the best lower bound known"); the bound is the Corollary of [Iw78], p. 226, $C(r)\ll r^2\log^2r$ for the longest run of consecutive integers each divisible by one of $r$ arbitrary primes, taken at $r=\pi(x)$ . Upper bounds: the trivial $S(k)\le k+1$ ; [Er76d]'s Rankin-type bound as printed; the site's $S(k)\ll k\log_3k/(\log_2k\log_4k)$ , deduced by the site from [FGKMT18] and recorded here as the site's; and the direct inversion of [FGKMT18]'s display (1.2), $S(k)\ll k\log_2k/(\log k\log_3k)$ , an authored one-line derivation recorded below and on its claim page (Ford Green Konyagin Maynard Tao, 2014), which is smaller than the site's display and implies it. This is a bounded negative finding, not a certificate of openness.

Er76d: Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Winnipeg, 1975), 25--44 (1976); the $L(n,k)$ passage on printed p. 26; the bibliography on p. 44. Library home: Erdos 1976; result page conjecture on p. 26.
FGKMT18: Ford, K., Green, B., Konyagin, S., Maynard, J. and Tao, T., Long gaps between primes. J. Amer. Math. Soc. 31 (2018), no. 1, 65--105, DOI 10.1090/jams/876; arXiv:1412.5029v3 (14 July 2016; the journal text not compared). Definition 1, Lemma 1.1 and (1.2), p. 3; (1.3) and the Iwaniec attestation, p. 4. Library home: Ford et al. 2018; result pages Theorem 1, display (1.2) and Lemma 1.1 with (1.3).
Er79d: Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. 33 (1979), 71--80; the $B(n)$ passage on printed p. 79. Library home: Erdos 1979; result page Section 3.
Er80: Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; the $f(x)$ passage on printed p. 106. Library home: Erdos 1980.
Iw78: Iwaniec, H., On the problem of Jacobsthal. Demonstratio Math. 11 (1978), no. 1, 225--231, DOI 10.1515/dema-1978-0121 (the printed pages; the Crossref record's 225--232 counts the blank page after the article). The definition of $C(r)$ on printed p. 225 and the Theorem and Corollary on p. 226. Library home: Iwaniec 1978; result pages Theorem and Corollary.
HaRi74: Halberstam, H. and Richert, H.-E., Sieve Methods. Academic Press (1974); [Er76d]'s reference [13], cited there for Rosser's bound without a page. Not held.
Ra38: Rankin, R. A., The difference between consecutive prime numbers. J. London Math. Soc. 13 (1938), 242--247; [Er76d]'s reference [15]. Not held (closed access; no request made).

Problem 929

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Proof claims

Discussion