文档介绍：BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 43, Number 1, Pages 3–23
S 0273-0979(05)01086-4
Article electronically published on October 6, 2005
THE GREEN-TAO THEOREM ON ARITHMETIC
PROGRESSIONS IN THE PRIMES:
AN ERGODIC POINT OF VIEW
BRYNA KRA
Abstract. A long-standing and almost folkloric conjecture is that the primes
contain arbitrarily long arithmetic progressions. Until recently, the only prog-
ress on this conjecture was due to van der Corput, who showed in 1939 that
there are infinitely many triples of primes in arithmetic progression. In an
amazing fusion of methods from analytic number theory and ergodic theory,
Ben Green and Terence Tao showed that for any positive integer k,there
exist infinitely many arithmetic progressions of length k consisting only of
prime numbers. This is an introduction to some of the ideas in the proof,
concentrating on the connections to ergodic theory.
1. Background
For hundreds of years, mathematicians have made conjectures about patterns
in the primes: one of the simplest to state is that the primes contain arbitrarily
long arithmetic progressions. It is not clear exactly when this conjecture was first
formalized, but as early as 1770 Lagrange and Waring studied the problem of how
large mon difference of an arithmetic progression of k primes must be. A
natural extension of this question is to ask if the prime numbers contain arbitrarily
long arithmetic progressions.
Support for a positive answer to this question is provided by the following simple
heuristic. The Prime Number Theorem states that the number of prime numbers
less than the integer N is asymptotically N/ log N. It follows that the density
of primes around a positive large x ∈ R is about 1/ log
sequence of prime numbers in {1,...,N} by choosing integers at random, with an
integer in {1,...,N} being chosen with probability 1/ log N, then there ought to be
approximately N 2/ logk N prog