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Claim. Assume the Hardy–Littlewood prime tuples conjecture in the uniform form stated below. Theorem 1 of P. X. Gallagher, On the distribution of primes in short intervals, then gives Poisson statistics for primes in short intervals: for every fixed $\lambda>0$ and every fixed integer $k\ge0$, the number of integers $n\le N$ for which the interval $(n,n+\lambda\log N]$ contains exactly $k$ primes is asymptotic to $e^{-\lambda}\lambda^k/k!$ times $N$. Banks's paper on ratios of consecutive prime gaps (card, Section 2.2) records the consequence for the gaps $d_n=p_{n+1}-p_n$ that Gallagher derives from it: under the same hypothesis, for every fixed $c\ge0$ the proportion of $n\le N$ with $d_n>c\log p_n$ tends to $e^{-c}$. Since $\log p_n/\log n\to1$, for every $\varepsilon>0$ and all large $n$ the inequality $d_n<c\log n$ implies $d_n<c(1+\varepsilon)\log p_n$ and is implied by $d_n<c(1-\varepsilon)\log p_n$, so the density of the $n$ with $(p_{n+1}-p_n)/\log n<c$ lies between $1-e^{-c(1-\varepsilon)}$ and $1-e^{-c(1+\varepsilon)}$ for every $\varepsilon>0$; letting $\varepsilon\to0$, the density $f(c)$ of Problem 234 exists for every $c\ge0$ and equals $1-e^{-c}$, a continuous function of $c$, which is both assertions of the problem. The passage from $\log p_n$ to $\log n$ is this corpus's own one-line deduction, not a statement of either paper. Tao states the same consequence on the site's discussion thread (29 September 2025): on the prime tuples conjecture the normalized gaps have an exponential distribution, so $f(c)=1-e^{-c}$.

Hypothesis. Gallagher's theorem assumes the Hardy–Littlewood asymptotic for prime tuples: for each fixed $k$, the number of $n\le N$ for which $n+d_1,\ldots,n+d_k$ are all prime is $(\mathfrak S(d_1,\ldots,d_k)+o(1)), N/(\log N)^k$, with $\mathfrak S$ the singular series, and the asymptotic is assumed to hold uniformly over distinct shifts $d_1,\ldots,d_k$ in $[1,h]$ with $h$ of order $\lambda\log N$. The proof averages the singular series over such shifts, where its mean is $1$, and reads off the Poisson moments. The hypothesis is unproved, and the claim gives no unconditional answer.

Scope. The claim is conditional and settles no standing of the problem by itself: unconditionally, neither the existence of $f(c)$ for any $c>0$ nor its continuity is known. A one-sided unconditional tail bound claimed in a 2026 manuscript is recorded on the problem page.

Acceptance. The result is refereed: P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), no. 1, 4--9, the paper link, with a corrigendum in Mathematika 28 (1981), 86. The site labels the problem OPEN and its commentary does not mention the result, so no curator acceptance is listed. The page is dated by the issue's publication month, June 1976, as the publisher's record gives it; the day in the page name is a placeholder.

Depends on. Nothing on this wiki beyond the cited paper; its hypothesis is stated above.