Teorema de Green-Tao

En teoría de números, el teorema de Green-Tao, demostrado por Ben Green y Terence Tao en 2004, establece que el sequence de número primo contiene progresión aritmética arbitrariamente largos. En otras palabras, para cada número natural k, existen arithmetic progressions of primes con términos k. La demostración es una extensión de Szemerédi's theorem. El problema se remonta a las investigaciones de Joseph-Louis Lagrange y Waring alrededor de 1770.​

Declaración

Sea $\pi (N)$ el número de números primos menores o iguales que $N$. Si $A$ es un subconjunto de los números primos tal que

$\limsup _{N\rightarrow \infty }{\dfrac {|A\cap [1,N]|}{\pi (N)}}>0$,

then for all positive número enteros $k$, the set $A$ contains infinitely many arithmetic progressions of length $k$. In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions.

In their later work on the generalized Número primo gemelo, Green and Tao stated and conditionally proved the asymptotic formula

$({\mathfrak {S}}_{k}+o(1)){\frac {N^{2}}{(\log N)^{k}}}$

for the number of k tuples of primes $p_{1}<p_{2}<\dotsb <p_{k}\leq N$ in arithmetic progression.​ Here, ${\mathfrak {S}}_{k}$ is the constant

${\mathfrak {S}}_{k}:={\frac {1}{2(k-1)}}\left(\prod _{p\leq k}{\frac {1}{p}}\left({\frac {p}{p-1}}\right)^{\!k-1}\right)\!\left(\prod _{p>k}\left(1-{\frac {k-1}{p}}\right)\!\left({\frac {p}{p-1}}\right)^{\!k-1}\right)\!.$

The result was made unconditional by Green–Tao​ and Green–Tao–Ziegler.​

Overview of the proof

Green and Tao's proof has three main components:

1. Szemerédi's theorem, 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.
2. 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.
3. 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 diferencia entre dos números primos consecutivoss.​ Once the pseudorandomness of the set is established, the transference principle may be applied, completing the proof.

Numerous simplifications to the argument in the original paper​ have been found. Conlon, Fox y 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 números primos en progresión aritmética:​

468,395,662,504,823 + 205,619 · 223,092,870 · n, for n= 0 to 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 primes:

6,171,054,912,832,631 + 366,384 · 23# · n, for n= 0 to 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 primes (sucesión A204189 en OEIS):

43,142,746,595,714,191 + 23,681,770 · 23# · n, for n= 0 to 25.

In September 2019 Rob Gahan and PrimeGrid found the first known case of 27 primes (sucesión A327760 en OEIS):

224,584,605,939,537,911 + 81,292,139 · 23# · n, for n= 0 to 26.

Extensions and generalizations

Many of the extensions of Szemerédi's theorem hold for the primes as well.

Independently, Tao and Ziegler​ y Cook, Magyar y Titichetrakun​​ derivaron una generalización multidimensional del teorema de Green-Tao. Fox y Zhao también simplificaron la prueba Tao-Ziegler.​

En 2006, Tao y Ziegler ampliaron el teorema de Green-Tao para cubrir progresiones polinómicas.​​ Más precisamente, dado cualquier polinomio de valores enteros P₁, ..., P_k en una m desconocida, todos con término constante 0, hay infinitos números enteros x, ' 'm tal que x + P₁(m), ..., x + Pk (m) son simultáneamente primos. El caso especial cuando los polinomios son m, 2m, ..., km implica el resultado anterior de que existen progresiones aritméticas de números primos de longitud k.

Tao demostró un análogo del teorema de Green-Tao para Entero gaussiano.​

Véase también

• Erdős conjecture on arithmetic progressions
• Teorema de Dirichlet sobre progresiones aritméticas
• Arithmetic combinatorics

Lectura adicional

• Conlon, David; Fox, Jacob; Zhao, Yufei (2014). «The Green–Tao theorem: an exposition». EMS Surveys in Mathematical Sciences 1 (2): 249-282. MR 3285854. S2CID 119301206. arXiv:1403.2957. doi:10.4171/EMSS/6. 
• Gowers, Timothy (2010). «Decompositions, approximate structure, transference, and the Hahn–Banach theorem». London Mathematical Society 42 (4): 573-606. MR 2669681. S2CID 17216784. arXiv:0811.3103. doi:10.1112/blms/bdq018. 
• Green, Ben (2007). «Long arithmetic progressions of primes». En Duke, William; Tschinkel, Yuri, eds. Analytic number theory. Clay Mathematics Proceeding 7. Providence, RI: American Mathematical Society. pp. 149-167. ISBN 978-0-8218-4307-9. MR 2362199. 
• Host, Bernard (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)]. Astérisque (en francés) (307): 229-246. Bibcode:2006math......9795H. MR 2296420. arXiv:math/0609795. 
• Kra, Bryna (2006). «The Green–Tao theorem on arithmetic progressions in the primes: an ergodic point of view». Bulletin of the American Mathematical Society 43 (1): 3-23. MR 2188173. doi:10.1090/S0273-0979-05-01086-4. 
• Tao, Terence (2006). «Arithmetic progressions and the primes». Collectanea Mathematica. Extra: 37-88. MR 2264205. Archivado desde el original el 5 de agosto de 2015. Consultado el 5 de junio de 2015. 
• Tao, Terence (2006). «Obstructions to uniformity and arithmetic patterns in the primes». Pure and Applied Mathematics Quarterly 2 (2): 395-433. MR 2251475. S2CID 6939076. arXiv:math/0505402. doi:10.4310/PAMQ.2006.v2.n2.a2. 
• Tao, Terence (7 de enero de 2008). «AMS lecture: Structure and randomness in the prime numbers». 

• Datos: Q922012