Problem 238

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claims/: The 1 claim page of Problem 238, one per claimant's result; the problem's standing derives from them.

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Statement. Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the difference between any two is $>c_2$?

Status. Open.

Source. erdosproblems.com/238, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #238, https://www.erdosproblems.com/238.

References.

• [Er49c] Erdős, P., On some applications of Brun's method. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57-63.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation (last edited 16 July 2026) fixes $c_1,c_2>0$ and asks whether every sufficiently large $x$ admits more than $c_1\log x$ consecutive primes below $x$ with all pairwise differences above $c_2$. The site labels the problem OPEN, and its commentary credits Erdős [Er49c] with the case of small $c_1$: for every $c_2>0$ the answer is yes once $c_1$ is small enough in terms of $c_2$ (Theorem 3 of that paper, proved by Brun's method through Schnirelmann's bound on the number of small prime gaps). That result is recorded in claims/ as an accepted partial claim with refereed evidence; the question for every pair $c_1,c_2>0$, in particular for $c_1$ large, is not answered by it, and the derived standing stays open. The site's thread discusses a conditional route through a uniform form of the Hardy–Littlewood prime tuples conjecture, but no proof or disproof of the full question is claimed there. The only formalization recorded is the formal-conjectures statement named under Formalization. This page records no literature search beyond the site and the cited paper.

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_1949_applications_brun_s_method
• erdos_1949_applications_brun_s_method / theorem_3