Erdős Problem #962

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.

A witness consists of a positive integer starting point $m\leq n$ and a length $k\geq1$. Each of $m+1,\ldots,m+k$ must have a prime divisor greater than this same length $k$. Different terms may use different prime divisors. The condition involves the greatest prime divisor of each term, not a requirement that all its prime factors exceed $k$. [S1] [S2]

Statement recorded in the supplied source.

Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers $ m+1,\ldots,m+k $ are divisible by at least one prime $>k$. Estimate $k(n)$. [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Close the gap between exp(Theta(sqrt(log n log log n))) and the near-square-root upper bound, and reconcile the precise priority and constants in the 1976 and Tang lower bounds. [S1]

Short English statement

How long a block beginning before a given bound can have every term divisible by a prime larger than the block length?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2412 (EP-962), 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 962, https://www.erdosproblems.com/962. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 P. Erdős, Problems and results on consecutive integers, Publ. Math. Debrecen 23 (1976), 271-282. https://publi.math.unideb.hu/paper/3385. Bibliographic information from the supplied record.

• S4 Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. Bibliographic information from the supplied record.