---
title: Small gaps between primes or almost primes
url: https://www.emergentmind.com/papers/0506067
type: paper
arxiv_id: '0506067'
arxiv_url: https://arxiv.org/abs/0506067
published: '2005-06-03'
categories:
- math.NT
---

# Small gaps between primes or almost primes

## Abstract

Let $p_n$ denote the $n^{th}$ prime. Goldston, Pintz, and Yildirim recently proved that $ \liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0.$ We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that $\liminf_{n\to \infty} (q_{n+1}-q_n) \le 26.$ If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.