Very small intervals containing at least three primes

TL;DR

This paper proves that a positive proportion of intervals between twice consecutive primes contain at least three primes, using a Cramér-like probabilistic model to analyze prime distribution.

Contribution

It establishes that intervals of the form (2p_n, 2p_{n+1}) containing at least three primes have positive density, a new result in prime interval analysis.

Findings

Intervals of the form (2p_n, 2p_{n+1}) contain at least three primes with positive density.

Uses a Cramér-like model to analyze prime distribution in specific intervals.

Provides probabilistic evidence for the frequency of multiple primes in small intervals.

Abstract

Let $p_n$ is the $n$-th prime. With help of the Cram\'er-like model, we prove that the set of intervals of the form $(2p_n,\enskip2p_{n+1})$ containing at list 3 primes has a positive density with respect to the set of all intervals of such form.