Primes in tuples I

By D. Goldston, Janos Pintz, C. Yıldırım. 2005

283 citations recorded by the source corpus.

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 Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, lim inf n→∞ Pn+1-Pn/log Pn/log = 0. We will quantify this result further in a later paper.

Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/efc95b2b0e383fe855832afd46ba973305035999

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In Algebra and Number Theory

In Geometry and Topology

In Theoretical Computer Science

Citations in the corpus

Small gaps between primes (2013, methodology): “The key new idea in the paper of Goldston, Pintz and Yıldırım [5] was to consider more general sieve weights of the form for a suitable smooth function F .”
Variants of the Selberg sieve, and bounded intervals containing many primes (2014, methodology): “Unconditionally, we have the following claims: (i) DHL[50; 2].”
Primes in tuples II (2007, methodology): “In [7] we proved that if the primes satisfy a level of distribution θ> 12 then there is an absolute constant M(θ) for which pn+1−pn 6M(θ) for infinitely many n.”
Bounded gaps between primes (2014, methodology): “We adopt the following notation of [6].”
Open problems in additive combinatorics (2007, unknown): “We tried to include primarily those problems we came across in our research, or at least lying close to the area of our research interests.”
Small gaps between products of two primes (2006, methodology): “†Chen’s result was announced in 1966 [3].”