Erdős Problem #12

The problem

Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with

$$\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0?$$

Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with

$$\lvert A\cap\{1,\ldots,N\}\rvert<N^{1-c}?$$

Is it true that

$$\sum_{n\in A}\frac{1}{n}<\infty?$$

Statement quoted from erdosproblems.com/12, maintained by Thomas Bloom. Original source: [ErSa70][Er73][Er75b][Er77c][Er80,p.113][Er92c][Er95c][Er97][Er97b][Er97e][Er98]. 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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