Green–Tao theorem

Green–Tao theorem

In mathematics, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, [Ben Green and Terence Tao, [http://arxiv.org/abs/math.NT/0404188 The primes contain arbitrarily long arithmetic progressions] ,8 Apr 2004.] states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for any natural number "k", there exist "k"-term arithmetic progressions of primes. The proof is an extension of Szemerédi's theorem.

In 2006, Tao and Tamar Ziegler extended the result to cover polynomial progressions. [Terence Tao, Tamar Ziegler, [http://arXiv.org/abs/math.NT/0610050 The primes contain arbitrarily long polynomial progressions] ] More precisely, given any integer-valued polynomials "P"1,..., "P""k" in one unknown "m" with vanishing constant terms, there are infinitely many integers "x", "m" such that "x" + "P"1("m"), ..., "x" + "P""k"("m") are simultaneously prime. The special case when the polynomials are "m", 2"m", ..., "km" implies the previous result that there are length "k" arithmetic progressions of primes.

These results were existence theorems and did not show how to find the progressions. On January 18, 2007, Jaroslaw Wroblewski found the first known case of 24 primes in arithmetic progression: [Jens Kruse Andersen, [http://hjem.get2net.dk/jka/math/aprecords.htm Primes in Arithmetic Progression Records] . Retrieved on 2008-09-08] :468395662504823 + 205619 × 23# × "n", for "n" = 0 to 23 (23# = 223092870).On May 17, 2008, Wroblewski and Raanan Chermoni found the first known case of 25 primes::6171054912832631 + 366384 × 23# × "n", for "n" = 0 to 24.

ee also

*Szemerédi's theorem
*Erdős conjecture on arithmetic progressions
*Dirichlet's theorem on arithmetic progressions
*Arithmetic combinatorics

References

External links

* [http://mathworld.wolfram.com/news/2004-04-12/primeprogressions MathWorld news article on proof]


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Terence Tao — Infobox Scientist name = Terence Chi Shen Tao caption = Terence Tao in March, 2006 at the Erdos Memorial Conference in Memphis, TN birth date = birth date and age|1975|07|17 residence = Los Angeles, California nationality = Australian field =… …   Wikipedia

  • Dirichlet's theorem on arithmetic progressions — In number theory, Dirichlet s theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other… …   Wikipedia

  • Térence Tao — Terence Tao Terence Tao Terence Tao né le 17 juillet 1975 à Adelaide en Australie, est un mathématicien qui travaille principalement dans les domaines de l analyse harmonique, les équations aux dérivées partielles, la combinatoire, la théorie ana …   Wikipédia en Français

  • Ben J. Green — Infobox Scientist box width = name = Ben J. Green image width = caption = birth date = birth date and age|1977|02|27 birth place = Bristol, England death date = death place = residence = Cambridge, England citizenship = nationality = British… …   Wikipedia

  • Prime number theorem — PNT redirects here. For other uses, see PNT (disambiguation). In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are… …   Wikipedia

  • Szemerédi's theorem — In number theory Szemerédi s theorem refers to the proof of the Erdős–Turán conjecture. In 1936 Erdős and Turan conjecturedcitation|authorlink1=Paul Erdős|first1=Paul|last1=Erdős|authorlink2=Paul Turán|first2=Paul|last2=Turán|title=On some… …   Wikipedia

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… …   Wikipedia

  • Analytic number theory — In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve number theoretical problems. [Page 7 of Apostol 1976] It is often said to have begun with Dirichlet s introduction of… …   Wikipedia

  • List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

  • List of unsolved problems in mathematics — This article lists some unsolved problems in mathematics. See individual articles for details and sources. Contents 1 Millennium Prize Problems 2 Other still unsolved problems 2.1 Additive number theory …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”