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**The primes contain arbitrarily long arithmetic progressions.**
*(English)*
Zbl 1191.11025

This paper needs little introduction: in 2004, the authors proved [arxiv:math/0404188] that the primes contain arbitrarily long arithmetic progressions, a startling result considering that the previous state of the art had been an infinitude of four term arithmetic progressions in which three elements were prime and the fourth a product of at most two primes. As frequently happens when an old problem falls, the solution also precipitated a vast new theory of linear forms in the primes which looks like it will lead to a resolution of the Hardy-Littlewood conjecture for essentially all systems except those describing structures such as twin primes or the Goldbach conjecture. This will be a major achievement, and although the theory has moved on somewhat from this opening of the door, it is still very much worth reading.

The main idea of the paper is a transference principle allowing the authors to transfer results from vanilla structures to pseudo-random versions. They then use some estimates of Goldston and Yıldırım to show that in a suitable sense subsets of the primes behave pseudo-randomly, which allows them to transfer Szemerédi’s theorem to subsets of the primes. A good introduction to this sphere of ideas may be found in the earlier paper of B. Green [Ann. Math. (2) 161, No. 3, 1609–1636 (2005; Zbl 1160.11307)] where this result is proved for three term progressions.

The main idea of the paper is a transference principle allowing the authors to transfer results from vanilla structures to pseudo-random versions. They then use some estimates of Goldston and Yıldırım to show that in a suitable sense subsets of the primes behave pseudo-randomly, which allows them to transfer Szemerédi’s theorem to subsets of the primes. A good introduction to this sphere of ideas may be found in the earlier paper of B. Green [Ann. Math. (2) 161, No. 3, 1609–1636 (2005; Zbl 1160.11307)] where this result is proved for three term progressions.

Reviewer: Tom Sanders (Cambridge)

### MSC:

11B25 | Arithmetic progressions |

11N13 | Primes in congruence classes |

37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |

11A41 | Primes |

### Citations:

Zbl 1160.11307
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\textit{B. Green} and \textit{T. Tao}, Ann. Math. (2) 167, No. 2, 481--547 (2008; Zbl 1191.11025)

### Online Encyclopedia of Integer Sequences:

Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.Initial prime in set of 4 consecutive primes with common difference 6.

a(n) = the least positive d such that for p=prime(n), the numbers p+0d, p+1d, p+2d, ..., p+(p-1)d are all primes.

Least number which is the end of an arithmetic progression of n numbers that are the sums of two nonzero squares.

Initial terms of arithmetic progression of primes in A005115 with duplicates removed.

Numbers k such that 3*k-4 and 2^k-1 are prime.

Numbers k such that 56211383760397 + 44546738095860*k is prime.

Numbers k such that 11410337850553 + 4609098694200*k is prime.

Semiprimes of the form 3*n*2^n - 3*n - 2^(2+n) + 4.

a(n) is the largest number in an n-term AP of Chen primes.

Length of the longest sequence of equidistant primes among the first n primes.

Number of prime quadruples p < q < r < s in arithmetic progression with all members less than or equal to prime(n).