# Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim

@article{Soundararajan2006SmallGB, title={Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim}, author={Kannan Soundararajan}, journal={Bulletin of the American Mathematical Society}, year={2006}, volume={44}, pages={1-18} }

In early 2005, Dan Goldston, Janos Pintz, and Cem Yildirim [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes. Before their work, it was known only that there were infinitely many gaps which were about a quarter the size of the average gap. The new result may be viewed as a…

## 11 Citations

Primes are No Longer Lonely: A Tale of Twin Primes and the Devoted Chinese Mathematician Yitang Zhang

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What is mathematics? Kronecker said: “God made the integers, all the rest is the work of man.” What makes integers? Prime numbers! Indeed, every integer can be written, essentially uniquely, as a…

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A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of…

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The aim of this expository article is to explain recent ideas in sieve theory which have been applied to prove results about prime gaps. We start by discussing the sieve of Erastosthenes, Brun’s pure…

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A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of…

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α e−t dt as x→∞, for any two fixed real numbers β > α ≥ 0. Gallagher’s calculation [Ga] shows that this conjecture can be deduced from the Hardy–Littlewood prime k-tuples conjecture (see [S2]). Hence…

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We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the…

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We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the…

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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…

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This essay discusses the method introduced by James Maynard in 2013 to show boundedness of gaps between primes. Also, two subsequent improvements to this method by the Polymath8b group are…