Problem 961

Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$ . Estimate $f(k)$ .

Source: erdosproblems.com/961

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem OPEN (page last edited 3 April 2026). No source determines the order of $f(k)$ ; what is in hand is a two-sided bound with a large gap. Upper bounds: $f(k)\le k$ (Sylvester--Schur, reproved by Erdős in 1934; claim page (Erdős, 1934)), $f(k)\le c_1k/\log k$ (Erdős 1955, Theorem 1; Erdős's 1976 survey restates it as $f(k)<3k/\log k$ ; claim page (Erdős, 1955)), and $f(k)<c_1k\log\log\log k/(\log k\log\log k)$ , the bound the site credits to Jutila [Ju74] and Ramachandra and Shorey [RaSh73], whose papers are not held here: it is quoted from Erdős's 1976 survey, which attributes it jointly to Jutila, Ramachandra and Shorey, and from the site (claim page (Jutila, Ramachandra and Shorey, 1973)). Each bound is an accepted partial claim on its journal publication. Lower bound: from Rankin's prime-gap theorem, Erdős's 1955 display (3), $f(k)>c_3\log k\log\log k\log\log\log\log k/(\log\log\log k)^2$ . Erdős expected $f(k)=o(k^\epsilon)$ and probably at most a power of $\log k$ (1976), and guessed $f(k)=(1+o(1))(\log k)^2$ on Cramér's conjecture (1955). No later source improving either bound was found in the search whose scope the Current assessment records; this is a bounded negative finding, not a certificate of openness.

Er76e: Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen 23 (1976), no. 3--4, 271--282, DOI 10.5486/pmd.1976.23.3-4.15 (Crossref record); the definition of $f(k)$ , the two classical bounds and display (1) on printed p. 271; the reference list, printed p. 282. Library home: Erdos 1976; result page display (1).
Er34: Erdős, Paul, A theorem of Sylvester and Schur. J. London Math. Soc. 9 (1934), no. 4, 282--288, DOI 10.1112/jlms/s1-9.4.282 (Crossref record); the theorem, printed p. 282, its binomial form and lemma, p. 283. Library home: Erdős 1934; result page Erdős 1934, Theorem (Sylvester–Schur).
Er55d: Erdős, P., On consecutive integers. Nieuw Arch. Wisk. (3) 3 (1955), 124--128; Theorem 1 and displays (1)--(3), printed p. 124, display (4) and the small values, p. 125. Library home: Erdős 1955; result pages Theorem 1 and inequality (3).
Ju74: Jutila, Matti, On numbers with a large prime factor. II. J. Indian Math. Soc. (N.S.) 38 (1974), 125--130. Not held; no Crossref record (the journal carries no DOIs for that volume); the journal's current web host is listed under Search scope (below). Quoted second-hand from [Er76e] and the site.
RaSh73: Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. 24 (1973), no. 1, 99--111, DOI 10.4064/aa-24-1-99-111 (Crossref record). Not held; the EuDML record of the paper names a PDF at the ICM digital library. Quoted second-hand from [Er76e] and the site. [Er76e]'s reference [15] pairs these two papers with Shorey, T. N., On gaps between numbers with a large prime factor. II. Acta Arith. 25 (1974), no. 4, 365--373 (DOI 10.4064/aa-25-4-365-373), which the site does not cite.
vDTa26: van Doorn, W. and Tang, Q., Consecutive integers free of certain prime factors. arXiv:2606.19863v1 (18 June 2026), 5 pp. Adjacent context only: its $n_k$ concerns prime factors in $(k,2k)$ (Problem 451), not $f(k)$ . Library home: Doorn and Tang 2026.
OEIS: Sloane, N. J. A., Sequence A213253, The On-Line Encyclopedia of Integer Sequences (2012; entry last modified 28 June 2026, server time): $f(n)$ for $n\le268$ (b-file from Najman's paper, not consulted here); JSON record accessed.

Problem 961

Status

Proof claims

Discussion