{
  "id": "math/0606088",
  "version": "v2",
  "published": "2006-06-04T11:02:25.000Z",
  "updated": "2008-04-22T17:00:46.000Z",
  "title": "Linear Equations in Primes",
  "authors": [
    "Ben Green",
    "Terence Tao"
  ],
  "comment": "84 pages, numerous small changes made in the light of comments from the referees",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Consider a system \\Psi of non-constant affine-linear forms \\psi_1,...,\\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N -> \\infty, for the number of integer points n in K for which the integers \\psi_1(n),...,\\psi_t(n) are simultaneously prime. This implies many other well-known conjectures, such as the Hardy-Littlewood prime tuples conjecture, the twin prime conjecture, and the (weak) Goldbach conjecture. <p> In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms \\psi_1,...,\\psi_t are affinely related; this excludes the important ``binary'' cases such as the twin prime or Goldbach conjectures, but does allow one to count ``non-degenerate'' configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the Inverse Gowers-norm conjecture GI(s) and the Mobius and Nilsequences Conjecture MN(s), where s \\in {1,2,...} is the complexity of the system and measures the extent to which the forms \\psi_i depend on each other. For s = 1 these are essentially classical, and the authors recently resolved the cases s = 2.<p> Our results are therefore unconditional in the case s = 2, and in particular we can obtain the expected asymptotics for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally for any (non-degenerate) problem involving two linear equations in four prime unknowns.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-04-22T17:00:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear equations",
      "hardy-littlewood prime tuples conjecture",
      "goldbach conjecture",
      "inverse gowers-norm conjecture gi",
      "asymptotic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 84,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6088G"
    }
  }
}