Erdős Problem #488

The problem

Let $A$ be a finite set and

$$B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.$$

Is it true that, for every $m>n\geq \max(A)$,

$$\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?$$

Statement quoted from erdosproblems.com/488, maintained by Thomas Bloom. Original source: [Er61,p.236][Er66,p.150][Er80,p.112]. Current status there: open (falsifiable).

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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