Two weeks ago, yitang zhang announced his result establishing that bounded gaps between primes occur infinitely often, with the explicit upper bound of 70,000,000 given for this gap. Zhang to establish the existence of bounded gaps between primes. Which is currently the best result on bounded gaps between primes. Jun, 2018 in this work we prove this conjecture in a somewhat more general form. Due to further advances of maynard and tao and the collaborative polymath project, 70 million has been reduced to a few hundred. This work was partially supported by the jsps, kakenhi grant number 16k17574. The oldest known is according to bertrands postulat. In the rest of the thesis we will be mostly interested in the bounded gaps between primes. On small gaps between primes department of mathematics. Polymath explicit methods in number theory mathematisches forschungsinstitut oberwolfach july 10, 2015.
Bounded gaps between primes yitang zhang it is proved that liminf n. James maynard born 10 june 1987 is a british mathematician best known for his work on prime gaps. A major ingredient of the proof is a stronger version of the. Sep 20, 20 the main objectives of the polymath8 project, initiated by terry tao back in june, were to understand the recent breakthrough paper of yitang zhang establishing an infinite number of prime gaps bounded by a fixed constant, and then to lower that value of as much as possible. We propose a proof of the conjecture that every admissible prime ktuple matches infinitely many positions in the sequence of primes. Our method is a re nement of the recent work of goldston, pintz and yildirim on the small gaps between consecutive primes. Previous work by thorner has shown the existence of bounded gaps between primes in chebotarev sets 9. Frank thorne expanded on this result, proving bounded gaps in the set of squarefree numbers with r prime factors for any r. I think the argument for 246 shares more with maynards proof of bounded gaps which doesnt use the same innovations as zhang and only uses a little bit from zhangs proof. The polymath8 project was proposed to improve the bounds for small gaps between primes. On a conjecture of erdos, polya and turan on consecutive gaps. Small and large gaps between the primes there are many questions about the gaps between consecutive prime numbers which are not completely solved, even after decades of effort.
Bounded gaps between primes of a special form international. Or relatedly, there is an even number 2k between 2 and 246 for which there are infinitely many primes that differ by 2k. I recently watched a video about the recent breakthrough involving the gaps between primes. Recently, yitang zhang proved the existence of a nite bound bsuch that there are in nitely many pairs p n. Fast forward to 20 and we have an idea for the eighth polymath project. Bounded gaps between primes clay mathematics institute. Bounded gaps between primes yitang zhang abstract it is proved that liminf n. Its actually there are infinitely many primes that differ by up to 246.
Pdf expanding total sieve and patterns in primes semantic. The bounded gaps between primes polymath project a. The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. We wont discuss the polymath projects in between 1 and 8 for more details on these, see 5. In the present work we prove a common generalization of maynardtaos recent result about consecutive bounded gaps between primes and of the erdosrankin bound about large gaps between consecutive primes. Our proof is based on a method of banks, freiberg and maynard which is again based on the method of maynard, tao and the polymath 8 project which showed the existence of infinitely many prime gaps not exceeding 246. Various authors have adapted the maynard framework to establish bounded gaps between primes in distinguished subsets, such as primes in beatty sequences 1, and primes with a given artin symbol. Introduction by enrico bombieri, institute for advanced study, princeton, nj, usa.
The bounded gaps between primes polymath project a retrospective. This can be seen as a massive breakthrough on the subject of twin primes and other. Polymath is a proven computational system that has been specifically created for educational or professional use. This is the wiki for polymath projects massively collaborative online mathematical projects.
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. They proved this by sieving by extracting prime pairs and prime tuples from all integers. Since then there has been a flurry of activity in reducing this bound, with the current record being 4,802,222 but likely to improve at least by a little. On scaling up the polymath project michael nielsen, march 25, 2009. This paper became irrelevant due to a quick progress in the polymath project of terrence tao.
The focus of this research school was, primarily, to bring young researchers up to date with the problem of small gaps between primes and discuss some of the main tools used by the mathematicians who have worked on this problem in the last year or so. An absolute lower bound has already been given, and no absolute upper bound exists, as has been mentioned, but there exist relative upper bounds which are however fairly weak as you shall see. Which is currently the best improvement on this result. One of the oldest open problems in analytic number theory. Polymath explicit methods in number theory mathematisches forschungsinstitut oberwolfach july 6, 2015. The polymath8b project 8 has subsequently improved these bounds.
Zhang had just submitted a proof that there are infinitely many consecutive primes which differ by at most 70,000,000. Bounded gaps between products of primes with applications to ideal class groups and elliptic curves frank thorne abstract. A note on bounded gaps between primes harvard university. I have an idea that im sure is wrong, but i dont know why. Tao, and the polymath project 14, so that the best known bound on gaps between primes, at least at the time of writing, is 252. In the present paper, the author makes signi cant progress in the bounds of gaps between primes. For instance, the twin prime conjecture, which asserts that the gap between primes can equal 2 infinitely often, remains open. The idea of such projects originated in tim gowers blog post is massively collaborative mathematics possible many polymath projects will be proposed, planned, and run at this blog a polymath logo is being trialled.
Remarkably, the techniques of maynard and tao also enable one to achieve bounded gaps between mconsecutive primes, i. Bounded gaps between primes home the polymath8 project, led by the fields medalist dr. A refinement of his argument was given by the polymath project whose goal is to reduce the gap of 70,000,000 and did so to 4680. Since then there has been a flurry of activity in reducing this bound, with the current record being 4,802,222 but likely to improve at least by a little bit in the near future. Oct 06, 2014 we wont discuss the polymath projects in between 1 and 8 for more details on these, see 5.
The approach is based on a concept of an expanding total sieve, which is a generalized sieving model representing cancellation of arithmetic progressions of matching positions of a given admissible ktuple during the infinite process of supplementing the sieve. Last week i attended the bounded gaps between primes research school at oxford university. Monday, september 22, 2014 to friday, september 26, 2014. A major ingredient of the proof is a stronger version of the bombierivinogradov theorem that.
Bounded gaps between primes in chebotarev sets request pdf. Sep 30, 2014 in this article, we collect the perspectives of several of the participants to these polymath projects, in order to form a case study of online collaborative mathematical activity, and to speculate on the suitability of such an online model for other mathematical research projects. Polymath8a, bounded gaps between primes, was a project to improve the bound h h 1 on the least gap between consecutive primes that was attained infinitely often, by developing the techniques of yitang zhang. This implies that there are infinitely many gaps between consecutive primes of size at most. Bounded gaps between primes andrew granville abstract. In his paper, titled bounded gaps between primes and bearing his name alone, zhang attacked the problem by proving that the. Bounded gaps between primes by yitang zhang abstract it is proved that liminf n. A short note on gaps between powers of consecutive primes david lowryduda abstract. Which is currently the best result on bounded gaps between. Tim gowers and the polymaths ian douglas the telegraph. Hisanobu shinya, on the density of prime differences less than a given magnitude which satisfy a certain inequality, arxiv. If you have more suggestions, please add them to the logo page, or add to the. Gaps between primes in beatty sequences request pdf. They obtained this number by strenghening zhangs estimates for type i, type ii, and type iii.
Our method is a re nement of the recent work of goldston, pintz and y ld r m on the small gaps between consecutive primes. The proof is essentially elementary, relying only on the bombierivinogradov theorem. A gentle introduction to the polymath project jason dyer, march 25, 2009. In the recent papers 12, goldston, graham, pintz, and ld r my use a arviant of the selberg sieve to prove the existence of small gaps between e 2 numbers, that is, squarefree. In the second section we will discus the goldston, pintz and yildirim sieve, then we will give the complete conditional proof on the existence of a bounded gap between in nitely many consecutive primes. Previous work by thorner has shown the existence of bounded gaps between. In this work we prove this conjecture in a somewhat more general form.
It comes as no surprise then that a seminar given yesterday afternoon at harvard by yitang zhang of the university of new hampshire reporting on his new paper bounded gaps between primes attracted a diverse audience. Due to further advances of maynard and tao and the collaborative polymath project. Sutherland massachusetts institute of technology on behalf of d. Naturally, if we assume that the primes have a higher level of distribution,then we can obtain stronger results. After completing his bachelors and masters degrees at university of cambridge in 2009, maynard obtained his phd from university of oxford at balliol college in 20 under the supervision of roger heathbrown. In any case, you are certainly right that 246 is the result of. In their breakthrough paper in 2006, goldston, graham, pintz and y. Yitang zhang proved that there are infinitely many pairs of primes differing by at most 70 million. Various authors have adapted the maynard framework to establish bounded gaps between primes in distinguished subsets, such as primes in beatty sequences 1, and primes with a. Bounded gaps between primes the polymath8 project, led by the fields medalist dr. On the ratio of consecutive gaps between primes springerlink.
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