Erdős Problem #873

The problem

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

$$F(A,X,k)<X^\epsilon?$$

Statement quoted from erdosproblems.com/873, maintained by Thomas Bloom. Original source: [Er92c]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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