# The path to recent progress on small gaps between primes

@article{Goldston2007ThePT, title={The path to recent progress on small gaps between primes}, author={D. A. Goldston and Janos Pintz and C. Y. Yildirim}, journal={arXiv: Number Theory}, year={2007} }

We present the development of ideas which led to our recent find- ings about the existence of small gaps between primes.

## 13 Citations

### Strings of congruent primes in short intervals

- Mathematics, PhilosophyJ. Lond. Math. Soc.
- 2011

It is proved that if (q, a )=1, then there are infinitely many pairs pr,p r+1 such that pr ≡ pr-1 ≡ a mod q and pr+1 − pr <� log pr.

### TOWARDS A PROOF OF THE TWIN PRIME CONJECTURE 33

- Mathematics, Philosophy
- 2011

Prime numbers differing by 2 are called twin primes. The twin prime conjecture states that the number of twin primes is infinite. Many attempts to prove or disprove this 2300-year old conjecture have…

### An Overview of Sieve Methods

- Mathematics
- 2010

An overview of the power of Sieve methods in number theory meant for the non-specialist is provided.

### Levels of Distribution and the Affine Sieve

- MathematicsAnnales de la Faculté des sciences de Toulouse : Mathématiques
- 2014

This article is an expanded version of the author's lecture in the Basic Notions Seminar at Harvard, September 2013. Our goal is a brief and introductory exposition of aspects of two topics in sieve…

### Three topics in additive prime number theory

- Mathematics
- 2007

This is an expository article to accompany my two lectures at the CDM conference. I have used this an excuse to make public two sets of notes I had lying around, and also to put together a short…

### The Goldston-Pintz-Yildirim sieveand some applications

- Mathematics
- 2013

The twin prime conjecture - that there exist infinitely many pairs of "twin primes" p, p + 2 - is among the most famous problems in number theory. In 2005, Goldston, Pintz and Yildirim (GPY) made a…

### The Goldston-Pintz-Yıldırım sieve and some applications

- Mathematics
- 2013

The twin prime conjecture that there exist infinitely many pairs of “twin primes” p, p + 2 is among the most famous problems in number theory. In 2005, Goldston, Pintz and Yıldırım (GPY) made a major…

### Small gaps between primes

- Mathematics
- 2007

AbstractCombining Goldston-Yildirim’s method on k-correlations of the truncated von Mangoldt function with Maier’s matrix method, we show that $$
\Xi _r : = \lim \inf _{n \to \infty } \tfrac{{p_{n +…

### Towards a Proof of the Twin Prime Conjecture

- Mathematics
- 2018

Introduction: All primes greater than or equal to five are of the form 6j –1 or 6j +1. m =1 to n ∏ Pm = Jn is the product of the first n primes. The number of (6j –1, 6j +1) pairs with no factor less…

### Distribution of prime numbers by the modified chi-square function

- Mathematics
- 2015

The statistical distribution of prime numbers represents an open problem in number theory still nowadays. The methodology of experimental mathematics has not yet been attempted in this field, thus…

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Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some…

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