The Primes Contain Arbitrarily Long Arithmetic Progressions in Annals of Mathematics 167 No. 2 pp. 481-547, March 2008
Princeton University Press, 2008. 1st Edition. FIRST EDITION IN ORIGINAL WRAPPERS OF THE CELEBRATED GREEN-TAO THEOREM. The theorem states that the prime numbers contain arbitrary long arithmetic progressions. For example, 5, 11, 17, 23, 29 is a sequence of five primes equally spaced, and so in arithmetic progression, the Green-Tao theorem says that you can find sequences of equally spaced primes which are as long as you like, though the spacing between them might be bigger” (IAS, Green-Tao Theorem).
A long-held and almost folkloric conjecture was that primes contain arbitrarily long arithmetic progressions. The problem can be traced back to investigations of Lagrange and Waring from around 1770. Green and Tao prove “there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi’s theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemeredi’s theorem that any subset of a sufficiently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Y ld r m, which we reproduce here. Using this, one may place (a large fraction of) the primes inside a pseudorandom set of “almost primes” (or more precisely, a pseudorandom measure concentrated on almost primes) with positive relative density” (Abstract, Green and Tao, The Primes Contain, Ann Math, p. 481, 2008).
ALSO INCLUDED: Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, pp. 325-348 by Kuo-Chang Chen; Toward a theory of rank one attractors, pp. 349-480 by Qiudong Wang, Lai-Sang Young; Cyclic homology, cdh-cohomology and negative K-theory, pp. 549-573 by Guillermo Cortinas, Christian Haesemeyer, Marco Schlichting, Charles Weibel; The Poincare inequality is an open ended condition, pp. 575-599 by Stephen Keith, Xiao Zhong; Growth and generation pp. 601-623 by Harald A. Helfgott; Uniform expansion bounds for Cayley graphs pp. 625-642 by Jean Bourgain, Alex Gamburd; Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents, pp. 643-680 by Marcelo Viana. Item #1402
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