Problem 1138

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}?

Source: erdosproblems.com/1138

Number theory
Primes

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SolvedDisproved

An accepted solution exists. The statement is false.

The site labels the problem DISPROVED (LEAN) (page last edited 06 July 2026) and credits Sunder, Kumrawat and Cheri, working with GPT 5.5, for the disproof: under the reading in the Formulation, two constants less than $1/2$ apart cannot both satisfy the asymptotic, so it fails for some $C>1$ . Whether it fails for every single $C>1$ is open. The paper, the credit and the Lean proofs of the disproof, not built here, are on the claim page (Sunder, Kumrawat and Cheri, 2026).

Problem 1138

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