Erdős Problem #1138

The problem

Let $x/2<y<x$ and $C>1$. If $d=\max_{p_n<x} (p_{n+1}-p_n)$, where $p_n$ denotes the $n$th prime, then is it true that

$$\pi(y+Cd)-\pi(y)\sim\frac{Cd}{\log y}?$$

Statement quoted from erdosproblems.com/1138, maintained by Thomas Bloom. Original source: [Va99,1.3]. Current status there: disproved (Lean).

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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