Acta Mathematica

, Volume 201, Issue 2, pp 213–305 | Cite as

The primes contain arbitrarily long polynomial progressions

  • Terence TaoEmail author
  • Tamar Ziegler


We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P 1, …, P k  ∈ Z[m] in one unknown m with P 1(0) = … = P k (0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with \(1 \leqslant m \leqslant x^\varepsilon\), such that x + P 1(m), …, x + P k (m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P j  = (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.


Convex Body Schwarz Inequality Polynomial System Transference Principle Distinguished Node 
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Copyright information

© Institut Mittag-Leffler 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesU.S.A.
  2. 2.Department of MathematicsTechnion – Israel Institute of TechnologyHaifaIsrael

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