Problem 234

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claims/: The 1 claim page of Problem 234, one per claimant's result; the problem's standing derives from them.

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Statement. For every $c\geq 0$ the density $f(c)$ of integers for which

$$\frac{p_{n+1}-p_n}{\log n}< c$$

exists and is a continuous function of $c$.

Status. Open, the site's label.

Source. erdosproblems.com/234, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #234, https://www.erdosproblems.com/234.

Formalization. Statement in formal-conjectures.

Current assessment

Unassessed beyond what this section states: the site's label is recorded without a status search, and the literature on the distribution of the normalized gaps $(p_{n+1}-p_n)/\log n$ has not been compiled here. Two sources are recorded because they bear on the problem. The 2026 release manuscript Positive lower density of large prime gaps claims, as its Theorem 1.1, that for every fixed $C>0$ a positive proportion of the indices $n\le N$ have $p_{n+1}-p_n>C\log p_n$ once $N$ is large. The manuscript claims nothing about this problem; its own target is the question of Erdős and Prachar behind Problem 968, and its claim is recorded there. It addresses neither question the problem asks, whether $f(c)$ exists for every $c\ge0$ and whether it is continuous in $c$, so the manuscript is background here and has no claim page on this problem.

The one conditional result is an accepted conditional claim: Gallagher's 1976 theorem derives, from a Hardy–Littlewood prime tuples asymptotic holding uniformly over shifts of order $\log N$, Poisson statistics for primes in short intervals, from which $f(c)$ exists and equals $1-e^{-c}$ for every $c\ge0$; Tao states the same consequence on the site's discussion thread (29 September 2025). The hypothesis is unproved, so the claim settles no standing and the problem stays open.

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.

• openai_2026_positive_lower_density_large_prime_gaps
• openai_2026_positive_lower_density_large_prime_gaps / theorem_1_1