Polymath8: Bounded Gaps between Primes

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 k-tuple 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 k₀ values, for every admissible k₀-tuple, there are infinitely many positive integers n to shift the k₀-tuple such that each shifted k₀-tuple contains at least two primes, and thus the width H of the admissible k₀-tuple establishes the upper bound for the gap between primes.

Zhang initally showed any k₀ ≥ 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 k₀ values and smaller H(k₀) values for them.

Code & Data

• Here is the code K0Finder (Java, bash, and maple are required), and a k₀ Table, for optimal k₀ 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 k-tuple 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): 2067-2199. [DOI]

Polymath8a authors: Wouter Castryck, Etienne Fouvry, Gergely Harcos, Emmanuel Kowalski, Philippe Michel, Paul Nelson, Eytan Paldi, Janos Pintz, Andrew V. Sutherland, Terence Tao, Xiao-Feng 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): 1-83. [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, Xiao-Feng 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 k-Tuples.

– The theoretical progress can be found in the blog of Terence Tao.