Erdős Problem #852

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.

$p_j$ 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 $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n<x$ the numbers $d_n,d_{n+1},\ldots,d_{n+h(x)-1}$ are all distinct. Estimate $h(x)$. In particular, is it true that $ h(x) >(\log x)^c $ for some constant $c>0$, and $ h(x)=o(\log x)? $ [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Determine the correct order of h(x). [S1]

Transcription note. Only unambiguous escape/formatting damage and nonmathematical serialization debris were repaired in the displayed source statement.

Short English statement

How long a run of consecutive prime gaps can be entirely distinct, and how does that length compare with powers of a logarithm?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2341 (EP-852), 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 852, https://www.erdosproblems.com/852. Reference from the supplied record; the live problem page was not independently checked for this compilation.