Erdős Problem #708

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.

The intended extremal quantity is an upper bound on the number of selected factors needed. The source writes $|B|=g(n)$, whereas the usual minimization is over $|B|\leq g(n)$; exact-size padding additionally requires a sufficiently long interval. This wording difference is retained in the source statement rather than concealed. [S1] [S2]

Statement recorded in the supplied source.

Let $g(n)$ be minimal such that for any $A\subseteq [2,\infty)\cap \mathbb{N}$ with $\lvert A\rvert =n$ and any set $I$ of $\max(A)$ consecutive integers there exists some $B\subseteq I$ with $\lvert B\rvert=g(n)$ such that $ \prod_{a\in A} a \mid \prod_{b\in B}b. $ Is it true that $ g(n) \leq (2+o(1))n? $ Or perhaps even $g(n)\leq 2n$? [S1]

Scope and interpretation.

The record leaves an exact-size versus at-most-size issue. The source statement is preserved and the conventional bound interpretation is identified; this section does not certify the two literal formulations as equivalent in every boundary case. [S1]

Short English statement

Can a product of prescribed integers always divide the product of at most twice as many distinct integers chosen from any sufficiently long interval?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2271 (EP-708), 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 708, https://www.erdosproblems.com/708. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 P. Erdős and J. Surányi, Bemerkungen zu einer Aufgabe eines mathematischen Wettbewerbs, Mat. Lapok 10 (1959), 39-48. https://users.renyi.hu/~p_erdos/1959-07.pdf. Bibliographic information from the supplied record.

• S4 Erdös, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. Bibliographic information from the supplied record.