Yitang zhangs techniques for bounding the gaps between primes soon led to rapid progress by the polymath group, and a further innovation by james maynard. Two proofs spark a prime week for number theory science. Bounded gaps between primes by yitang zhang now available. Zhangs work also set off a flurry of followup activity in the mathematics community, including a collaboration of a dozen or so mathematicians called the polymath8 project, which is attempting to reduce the maximum separation between primes in zhangs proof to something less than 70 million. Stony brook mathematics colloquium video bounded gaps between primes yitang zhang university of new hampshireias date. Yitang zhang bounded gaps between primes 2014 youtube. This can be seen as a massive breakthrough on the subject of twin primes and other. In his paper, titled bounded gaps between primes and bearing his name alone, zhang attacked the problem by proving that the number of primes that are less than 70 million units apart is. We give a short introduction to the recent breakthrough theorem of yitang zhang that there are infinitely many pairs of distinct primes p, q with p. An unknown mathematician, yitang zhang, has revolutionized his field and helped move forward a 2,000yearold conjecture about prime numbers. Bounded gaps between primes yitang zhang abstract it is proved that liminf n. Philosophy behind yitang zhangs work on the twin primes. Bounded gaps between primes andrew granville abstract. A major ingredient of the proof is a stronger version of the bombierivinogradov theorem that.
Yitang zhang in celebration of yitang zhang s recent result, consider the gaps between one prime and the next. Bounded gaps between primes by yitang zhang abstract it is proved that liminf n. Aperiodic to complete the story started as a rumour report in primes gotta stick together and confirmed in primes really do stick together, here we report that annals of mathematics has posted the pdf of. All large primes have a last digit of 1, 3, 7, or 9, with a 25% proportion of primes having each of these digits. An expository essay article pdf available in surveys in mathematics and its applications 122017. Shiuyuen cheng for pointing out the paper of soundararajan cited in this article, prof. In a recent paper zha in the annals of mathematics, yitang zhang. They may not be twin primes, but at least they are 70millionth cousins. Yitang zhangs pursuit of beauty in math the new yorker. Has anybody read yitang zhangs paper on prime gaps. The big experts in the field had already tried to make this approach work, granville said. Lo yang for information about chengbiao pan quoted below, prof.
Pdf new developments on the twin prime problem and. Zhangs original bound here was 70 million, but it has since been cut down to 246 thanks to the efforts of james maynard and an online collaborative polymath project. Yitang zhang recently published a new attack on the twin primes conjecture. Yitang zhang and the twin prime conjecture centers on an exciting string of mathematical discoveries. Twin primes as of 2014, the largest known twin primes are 37568016956852666669 1 these enormous primes have 200700 digits each. Yitang zhang proves landmark theorem in distribution of. Within weeks word spread a littleknown mathematician, with. Zhang, yitang zhang, yi tang 1nhndm bounded gaps between primes. Why did seemingly nobody notice yitang zhang before the.
Hes not a known expert, but he succeeded where all the experts had failed. He was a math instructor, not a tenuretrack professor, at the university of new hampshire from 1999 to 20, teaching prealgebra and calculus yitang zhang on rat. How yitang tom zhang proved a theorem that had stumped. Last year, in a breakthrough work of yitang zhang, it was shown that there were infinitely many gaps between primes of bounded size. The pursuit of beauty in pure mathematics is a tenet. It began in april 20 when yitang tom zhang, a virtually unknown mathematician working as an adjunct professor at the university of new hampshire, submitted a paper to the annals of mathematics. Yuan wang for detailed information on results related to the twin prime conjecture, and prof. A year ago april, the editors of the annals of mathematics, a journal published by the institute and princeton university, received an email with a submission by an unknown mathematician. Recently, yitang zhang proved the existence of a nite bound bsuch that there are in nitely many pairs p n. Our method is a refinement of the recent work of goldston, pintz and y. Yitang zhang proves weak version of twin prime conjecture. A major ingredient of the proof is a stronger version of the bombierivinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose. His counterintuitive findings show that special pairs.
Zhang had made an important breakthrough in number theory by solving. Because zhang was a nobody at a relatively unknown researchwise us university. In april 20, a lecturer at the university of new hampshire submitted a paper to the annals of mathematics. The twin prime conjecture states that there are an infinite number of twin primes prime numbers where one prime is two greater than the other. While we still dont know if there are an infinite number of pairs of primes whose difference is 2, he showed that. Counting from infinity yitang zhang and the twin prime conjecture a film by george csicsery in april 20, a lecturer at the university of new hampshire submi ed a paper to the annals of mathema cs within weeks word spread.
This better approximation gives a prediction that the number of twin primes less than a quadrillion should be about 1. On april 17, a paper arrived in the inbox of annals of mathematics, one of the disciplines preeminent journals. On the same day, in a lecture at harvard university, yitang tom zhang of the university of new hampshire in durham sketched a proof that there are infinitely many prime numbers separated by less than 70 million. Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Yitang zhang is a mathematician tackling truly hard questions in number theory leading to the first finite bound on a gap between prime numbers and the best possible qualitative approximation of. Bounded gaps between primes yitang zhang it is proved that liminf n. Premiering saturday, may 6 at 1pm on thirteen, counting from infinity.
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. Mathematician yitang zhang, 2014 macarthur fellow youtube. Pdf conjecture of twin primes still unsolved problem in. All large odd numbers can be expressed as the sum of three primes. In may 20, zhang made a huge step towards proving the conjecture. We have completed our study of the paper bounded gaps between primes by yitang zhang.
So, if the twin prime conjecture is true, it means the gap doesnt get larger, because theres always neighbor primes with gap of just 2. Conjecture of twin primes still unsolved problem in number theory. Are there any experts here who can explain the proof. It began in april 20 when yitang tom zhang, a virtually. To complete the story started as a rumour report in primes gotta stick together and confirmed in primes really do stick together, here we report that annals of mathematics has posted the pdf of bounded gaps between primes by yitang zhang on its to appear in forthcoming.
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