Green–Tao theorem

Green–Tao theorem In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In altre parole, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.[1] Contenuti 1 Dichiarazione 2 Overview of the proof 3 Numerical work 4 Estensioni e generalizzazioni 5 Guarda anche 6 Riferimenti 7 Further reading Statement Let {stile di visualizzazione pi (N)} denote the number of primes less than or equal to {stile di visualizzazione N} . Se {stile di visualizzazione A} is a subset of the prime numbers such that {displaystyle limsup _{Nrightarrow infty }{dfrac {|Spesso [1,N]|}{pi (N)}}>0} , then for all positive integers {stile di visualizzazione k} , il set {stile di visualizzazione A} contains infinitely many arithmetic progressions of length {stile di visualizzazione k} . In particolare, the entire set of prime numbers contains arbitrarily long arithmetic progressions.

In their later work on the generalized Hardy–Littlewood conjecture, Green and Tao stated and conditionally proved the asymptotic formula {stile di visualizzazione ({mathfrak {S}}_{K}+o(1)){frac {N^{2}}{(registro n)^{K}}}} for the number of k tuples of primes {stile di visualizzazione p_{1}K}sinistra(1-{frac {k-1}{p}}Giusto)!sinistra({frac {p}{p-1}}Giusto)^{!k-1}Giusto)!.} The result was made unconditional by Green–Tao [3] and Green–Tao–Ziegler.[4] Overview of the proof Green and Tao's proof has three main components: Il teorema di Szemerédi, which asserts that subsets of the integers with positive upper density have arbitrarily long arithmetic progressions. It does not a priori apply to the primes because the primes have density zero in the integers. A transference principle that extends Szemerédi's theorem to subsets of the integers which are pseudorandom in a suitable sense. Such a result is now called a relative Szemerédi theorem. A pseudorandom subset of the integers containing the primes as a dense subset. To construct this set, Green and Tao used ideas from Goldston, Pintz, and Yıldırım's work on prime gaps.[5] Once the pseudorandomness of the set is established, the transference principle may be applied, completando la dimostrazione.

Numerous simplifications to the argument in the original paper[1] have been found. Conlon, Fox & Zhao (2014) provide a modern exposition of the proof.

Numerical work The proof of the Green–Tao theorem does not show how to find the arithmetic progressions of primes; it merely proves they exist. There has been separate computational work to find large arithmetic progressions in the primes.

The Green–Tao paper states 'At the time of writing the longest known arithmetic progression of primes is of length 23, and was found in 2004 by Markus Frind, Paul Underwood, and Paul Jobling: 56211383760397 + 44546738095860 · k; K = 0, 1, . . ., 22.'.

On January 18, 2007, Jarosław Wróblewski found the first known case of 24 primes in arithmetic progression:[6] 468,395,662,504,823 + 205,619 · 223,092,870 · n, per n = 0 a 23.

The constant 223,092,870 here is the product of the prime numbers up to 23, more compactly written 23# in Primorial notation.

On May 17, 2008, Wróblewski and Raanan Chermoni found the first known case of 25 primi: 6,171,054,912,832,631 + 366,384 · 23# · n, per n = 0 a 24.

On April 12, 2010, Benoît Perichon with software by Wróblewski and Geoff Reynolds in a distributed PrimeGrid project found the first known case of 26 primi (sequence A204189 in the OEIS): 43,142,746,595,714,191 + 23,681,770 · 23# · n, per n = 0 a 25.

In September 2019 Rob Gahan and PrimeGrid found the first known case of 27 primi (sequence A327760 in the OEIS): 224,584,605,939,537,911 + 81,292,139 · 23# · n, per n = 0 a 26. Extensions and generalizations Many of the extensions of Szemerédi's theorem hold for the primes as well.

Independently, Tao and Ziegler[7] and Cook, Magyar, and Titichetrakun[8][9] derived a multidimensional generalization of the Green–Tao theorem. The Tao–Ziegler proof was also simplified by Fox and Zhao.[10] In 2006, Tao and Ziegler extended the Green–Tao theorem to cover polynomial progressions.[11][12] Più precisamente, given any integer-valued polynomials P1, ..., Pk in one unknown m all with constant term 0, there are infinitely many integers x, m such that x + P1(m), ..., X + Pk(m) are simultaneously prime. The special case when the polynomials are m, 2m, ..., km implies the previous result that there are length k arithmetic progressions of primes.

Tao proved an analogue of the Green–Tao theorem for the Gaussian primes.[13] See also Erdős conjecture on arithmetic progressions Dirichlet's theorem on arithmetic progressions Arithmetic combinatorics References ^ Jump up to: a b Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annali di matematica. 167 (2): 481–547. arXiv:math.NT/0404188. doi:10.4007/annals.2008.167.481. SIG 2415379. S2CID 1883951.. ^ Verde, Ben; Tao, Terence (2010). "Linear equations in primes". Annali di matematica. 171 (3): 1753–1850. arXiv:math/0606088. doi:10.4007/annals.2010.171.1753. SIG 2680398. S2CID 119596965. ^ Verde, Ben; Tao, Terence (2012). "The Möbius function is strongly orthogonal to nilsequences". Annali di matematica. 175 (2): 541–566. arXiv:0807.1736. doi:10.4007/annals.2012.175.2.3. SIG 2877066. ^ Verde, Ben; Tao, Terence; Ziegler, Tamar (2012). "An inverse theorem for the Gowers {stile di visualizzazione U^{s+1}[N]} -norma". Annali di matematica. 172 (2): 1231–1372. arXiv:1009.3998. doi:10.4007/annals.2012.176.2.11. SIG 2950773. ^ Goldston, Daniel A.; Pintz, John; Yıldırım, Cem Y. (2009). "Primes in tuples. io". Annali di matematica. 170 (2): 819–862. arXiv:math/0508185. doi:10.4007/annals.2009.170.819. SIG 2552109. S2CID 1994756. ^ Andersen, Jens Kruse. "Primes in Arithmetic Progression Records". Recuperato 2015-06-27. ^ Tao, Terence; Ziegler, Tamar (2015). "A multi-dimensional Szemerédi theorem for the primes via a correspondence principle". Israel Journal of Mathematics. 207 (1): 203–228. arXiv:1306.2886. doi:10.1007/s11856-015-1157-9. SIG 3358045. S2CID 119685169. ^ Cook, Brian; Magyar, Ákos (2012). "Constellations in {displaystyle mathbb {P} ^{d}} ". International Mathematics Research Notices. 2012 (12): 2794–2816. doi:10.1093/imrn/rnr127. SIG 2942710. ^ Cook, Brian; Magyar, Ákos; Titichetrakun, Tatchai (2018). "A Multidimensional Szemerédi Theorem in the primes via Combinatorics". Annals of Combinatorics. 22 (4): 711–768. arXiv:1306.3025. doi:10.1007/s00026-018-0402-4. S2CID 126417608. ^ Fox, Giacobbe; Zhao, Yufei (2015). "A short proof of the multidimensional Szemerédi theorem in the primes". Giornale americano di matematica. 137 (4): 1139–1145. arXiv:1307.4679. doi:10.1353/ajm.2015.0028. SIG 3372317. S2CID 17336496. ^ Tao, Terence; Ziegler, Tamar (2008). "The primes contain arbitrarily long polynomial progressions". Giornale di matematica. 201 (2): 213–305. arXiv:math/0610050. doi:10.1007/s11511-008-0032-5. SIG 2461509. S2CID 119138411. ^ Tao, Terence; Ziegler, Tamar (2013). "Erratum to "The primes contain arbitrarily long polynomial progressions"". Giornale di matematica. 210 (2): 403–404. doi:10.1007/s11511-013-0097-7. SIG 3070570. ^ Tao, Terence (2006). "The Gaussian primes contain arbitrarily shaped constellations". Giornale di analisi matematica. 99 (1): 109–176. arXiv:math/0501314. doi:10.1007/BF02789444. SIG 2279549. S2CID 119664036. Further reading Conlon, Davide; Volpe, Giacobbe; Zhao, Yufei (2014). "The Green–Tao theorem: an exposition". EMS Surveys in Mathematical Sciences. 1 (2): 249–282. arXiv:1403.2957. doi:10.4171/EMSS/6. SIG 3285854. S2CID 119301206. Gower, Timoteo (2010). "Decompositions, approximate structure, transference, and the Hahn–Banach theorem". Bollettino della London Mathematical Society. 42 (4): 573–606. arXiv:0811.3103. doi:10.1112/blms/bdq018. SIG 2669681. S2CID 17216784. Verde, Ben (2007). "Long arithmetic progressions of primes". In Duke, William; Tschinkel, Yuri (eds.). Teoria analitica dei numeri. Clay Mathematics Proceeding. vol. 7. Provvidenza, RI: Società matematica americana. pp. 149–167. ISBN 978-0-8218-4307-9. SIG 2362199. Host, Bernardo (2006). "Progressions arithmétiques dans les nombres premiers (d'après B. Green et T. Tao)" [Arithmetical progressions in the primes (after B. Green and T. Tao)] (PDF). Asterisco (in francese) (307): 229–246. arXiv:math/0609795. Bibcode:2006math......9795H. SIG 2296420. Kra, Bryna (2006). "The Green–Tao theorem on arithmetic progressions in the primes: an ergodic point of view". Bollettino dell'American Mathematical Society . 43 (1): 3–23. doi:10.1090/S0273-0979-05-01086-4. SIG 2188173. Tao, Terence (2006). "Arithmetic progressions and the primes". Collectanea Mathematica. Extra: 37–88. SIG 2264205. Archiviato dall'originale in poi 2015-08-05. Recuperato 2015-06-05. Tao, Terence (2006). "Obstructions to uniformity and arithmetic patterns in the primes". Pure and Applied Mathematics Quarterly. 2 (2): 395–433. arXiv:math/0505402. doi:10.4310/PAMQ.2006.v2.n2.a2. SIG 2251475. S2CID 6939076. Tao, Terence (2008-01-07). "AMS lecture: Structure and randomness in the prime numbers". Categorie: Ramsey theoryAdditive combinatoricsAdditive number theoryTheorems about prime numbers