Erdős Problem #873

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.

Blocks are consecutive in the increasing enumeration of $A$, not necessarily intervals of integers. The limit concerns $X\to\infty$ after $A$, $\epsilon$ and the block length are fixed. No uniformity in $A$ beyond that stated in the source is silently imposed. [S1] [S2]

Statement recorded in the supplied source.

Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that $ [a_i,a_{i+1},\ldots,a_{i+k-1}] < X, $ where the left-hand side is the least common multiple. Is it true that, for every $\epsilon >0$, there exists some $k$ such that $ F(A,X,k)<X^\epsilon? $ [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Obtain exponents tending to zero as block length grows. [S1]

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

Short English statement

For every increasing integer sequence, does taking sufficiently long consecutive blocks make the count of blocks with small least common multiple subpower in the bound?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2356 (EP-873), 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 873, https://www.erdosproblems.com/873. Reference from the supplied record; the live problem page was not independently checked for this compilation.