Polymath8: Bounded Gaps between Primes

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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 pk.

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 cϖ, cδ, i.
  • Here is the code KTupleFinder (Java) for minimizing the width value H of an admissible k-tuple for a given k.
Publications

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

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

rnoti-jj-15-cov1– Media coverage: Notices of the AMSDer 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.


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Work with the Polymath project was featured by Notices of the AMS

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Category : News , Projects

Notices of the American Mathematical SocietyThe Polymath8 participants was featured on the June/July cover of Notices of the American Mathematical Society, the world’s most widely read mathematical journal.

The photographs on the cover portray thirteen of the fifteen people, including a team of top mathematicians, as being primary participants of the Polymath8 project that improved dramatically the bound of prime gaps, based on the stunning work of  Dr. Yitang Zhang.

Read more at Notices of the AMS.


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Work with the Polymath project was published on Algebra & Number Theory

Category : News , Projects

The work with the Polymath project was published on Algebra & Number Theory.

Abstract: We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese remainder theorem conditions, obtaining an exponent of distribution 1/2+7/300.


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Work with the Polymath project was published on Research in the Mathematical Sciences

Category : News , Projects

The work with the Polymath project was published on Research in the Mathematical Sciences.

Here is the list of Polymath8b authors (arranged in alphabetical order of surname): 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

Abstract: For any m \geq 1, let H_m denote the quantity \liminf_{n \to \infty} (p_{n+m}-p_n), where p_n is the n^{\text{th}} prime. A celebrated recent result of Zhang showed the finiteness of H_1, with the explicit bound H_1 \leq 70000000. This was then improved by us (the Polymath8 project) to H_1 \leq 4680, and then by Maynard to H_1 \leq 600, who also established for the first time a finiteness result for H_m for m \geq 2, and specifically that H_m \ll m^3 e^{4m}. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H_1 \leq 12, improving upon the previous bound H_1 \leq 16 of Goldston, Pintz, and Y{\i}ld{\i}r{\i}m, as well as the bound H_m \ll m^3 e^{2m}. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further, and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H_1 \leq 246 unconditionally, and H_1 \leq 6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture we show the stronger statement that for any admissible triple (h_1,h_2,h_3), there are infinitely many n for which at least two of n+h_1,n+h_2,n+h_3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds, or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the “parity problem” argument of Selberg to show that the H_1 \leq 6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound H_m \ll m e^{(4-\frac{28}{157})m}, or H_m \ll m e^{2m} under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H_m when m=2,3,4,5.


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Work with the Polymath project was reported by Der Spiegel

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Category : News , Projects

This work with the Polymath8 project was to find narrow gaps between primes.

derspiegel

Mal steuerte Xiao-Feng Xie einen Vorschlag bei, ein Robotikexperte aus Pittsburgh. Mal meldete sich Terence Tao aus Los Angeles, den einige für den brillantesten aller lebenden Mathematiker halten. Wieder andere Anregungen kamen von einem Anonymus, der sich nur als v08ltu zu erkennen gab.

Read more at Der Spiegel (web | pdf).