Green–Tao theorem

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.