Polymath8: Bounded Gaps between Primes
Category : Projects
The Polymath8 project, led by the Fields Medalist Dr. Terence Tao and in collaboration with a team of top mathematicians, was launched to optimize the records of the bounded gaps between primes based on the breakthrough work of “Bounded gaps between primes” by Dr. Yitang Zhang. He proved that there are infinitely many pairs of primes with a finite gap, and thus resolved a weak form of the twin prime conjecture.
Introduction


In number theory, an admissible ktuple is a set of k distinct integers that do not include the complete modulo set of residue classes (i.e. the values 0 through p – 1) of any prime p ≤ k.
Zhang showed that in some k_{0} values, for every admissible k_{0}tuple, there are infinitely many positive integers n to shift the k_{0}tuple such that each shifted k_{0}tuple contains at least two primes, and thus the width H of the admissible k_{0}tuple establishes the upper bound for the gap between primes.
Zhang initally showed any k_{0} ≥ 3,500,000 (which leads to H ≤ 70,000,000) is adequate for the bounding purpose. A polymath8 project was then started to find smaller k_{0} values and smaller H(k_{0}) values for them.
Code & Data


 Here is the code K0Finder (Java, bash, and maple are required), and a k_{0} Table, for optimal k_{0} of MPZ^{(i)}(ϖ, δ) in different settings of c_{ϖ}, c_{δ}, i.
 Here is the code KTupleFinder (Java) for minimizing the width value H of an admissible ktuple for a given k.
Publications


 D. H. J. Polymath, et al. New equidistribution estimates of Zhang type, and bounded gaps between primes. Algebra & Number Theory, 2014, 8(9): 20672199. [DOI]
Polymath8a authors: Wouter Castryck, Etienne Fouvry, Gergely Harcos, Emmanuel Kowalski, Philippe Michel, Paul Nelson, Eytan Paldi, Janos Pintz, Andrew V. Sutherland, Terence Tao, XiaoFeng Xie
 D. H. J. Polymath, et al. Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences, 2014, 1(12): 183. [DOI]
Polymath8b authors: Ignace Bogaert, Aubrey de Grey, Gergely Harcos, Emmanuel Kowalski, Philippe Michel, James Maynard, Paul Nelson, Pace Nielsen, Eytan Paldi, Andrew V. Sutherland, Terence Tao, XiaoFeng Xie
Other Information


– Media coverage: Notices of the AMS; Der Spiegel
– Polymath8 project for finding bounded intervals with primes.
– New bounds on gaps between primes by Andrew V. Sutherland.
– A new database has been set for collecting narrow admissible kTuples.
– The theoretical progress can be found in the blog of Terence Tao.