Erdős Problem #968

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

For an integer set $A$, its natural density is $\lim_{X\to\infty}|A\cap[1,X]|/X$, when this limit exists. Upper and lower densities use limit superior and limit inferior. A request for density is not automatically satisfied by positive upper density alone.

$p_j$ denotes the $j$th positive prime in increasing order, starting with $p_1=2$. The prime-counting function is $\pi(x)=\#\{p\leq x:p\text{ is prime}\}$. Consecutive primes are consecutive in this full sequence. [S1] [S2]

Statement recorded in the supplied source.

Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n<u_{n+1}$ have positive density? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove that above-average prime gaps occur for a positive density of indices, or otherwise directly establish positive density for u_n<u_(n+1). [S1]

Short English statement

Does the ratio of the nth prime to its index increase at a positive proportion of indices?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2414 (EP-968), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 968, https://www.erdosproblems.com/968. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 P. Erdős and K. Prachar, Sätze und Probleme über p_k/k, Abh. Math. Sem. Univ. Hamburg 26 (1961/62), 251-256. https://www.renyi.hu/~p_erdos/1961-21.pdf. Bibliographic information from the supplied record.

• S4 Erdős, P. and Prachar, K., Sätze und {P}robleme über {$p\sb{k}/k$}. Abh. Math. Sem. Univ. Hamburg (1961/62), 251--256. Bibliographic information from the supplied record.