Erdős Problem #218

The problem

Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.

Statement quoted from erdosproblems.com/218, maintained by Thomas Bloom. Original source: [Er55c][Er57][Er61][Er65b][Er85c]. 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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