In number theory, the Green–Tao theorem, proven by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k {\displaystyle k} , there exist arithmetic progressions of primes with k {\displaystyle 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.
Statement Let π ( N ) {\displaystyle \pi (N)} denote the number of primes less than or equal to N {\displaystyle N} . If A {\displaystyle A} is a subset of the prime numbers such that
lim sup N → ∞ | A ∩ [ 1 , N ] | π ( N ) > 0 , {\displaystyle \limsup _{N\rightarrow \infty }{\frac {|A\cap [1,N]|}{\pi (N)}}>0,}
then for all positive integers k {\displaystyle k} , the set A {\displaystyle A} contains infinitely many arithmetic progressions of length k {\displaystyle k} . In particular, 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
( S k + o ( 1 ) ) N 2 ( log N ) k {\displaystyle ({\mathfrak {S}}_{k}+o(1)){\frac {N^{2}}{(\log N)^{k}}}}
for the number of k tuples of primes p 1 < p 2 < ⋯ < p k ≤ N {\displaystyle p_{1}<p_{2}<\dotsb <p_{k}\leq N} in arithmetic progression. Here, S k {\displaystyle {\mathfrak {S}}_{k}} is the constant
S k := 1 2 ( k − 1 ) ( ∏ p ≤ k 1 p ( p p − 1 ) k − 1 ) ( ∏ p > k ( 1 − k − 1 p ) ( p p − 1 ) k − 1 ) . {\displaystyle {\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:
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. 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. 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 & Zhao (2014) provide a modern exposition of the proof.
Numerical work
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