The primes contain arbitrarily long polynomial progressions

By Terence Tao, T. Ziegler. 2006

153 citations recorded by the source corpus.

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P1, …, Pk ∈ Z[m] in one unknown m with P1(0) = … = Pk(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 + P1(m), …, x + Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pj = (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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

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Citations in the corpus

Boosting and Differential Privacy (2010, background): “(This can be viewed as an information-theoretic analogue of the “dense model theorems” of [15], [25], [21], [20].”
Open problems in additive combinatorics (2007, unknown): “We tried to include primarily those problems we came across in our research, or at least lying close to the area of our research interests.”
The Complexity of Differential Privacy (2017, background): “When considering a single pair of random variables (Y,Z), the dense model theorem of [55, 100, 90] says that pseudodensity is equivalent to Y being computationally indistinguishable from a random variable Ỹ that truly has density at least ρ in Z.”
Decompositions, approximate structure, transference, and the Hahn–Banach theorem (2008, background): “We shall give a simple proof of a result that is implicit in their paper, and made explicit in a later paper of Tao and Ziegler, and then we shall explain informally how they use this result to prove their famous theorem.”
Some open problems on multiple ergodic averages (2011, methodology): “143, 145, 150, 163, 173, 178, 186, 193, 200, 208, 211, 212, 224, 229, 236, 237, 239, 245, 247, 254, 261, 287, 290]. Other sequences (rational, smooth, random, prime numbers, generalized polynomials): [44, 45, 50, 51, 55, 61, 62, 66, 67, 74, 78, 79, 91, 116, 120, 123, 132, 133, 138, 139, 146, 147, 14”
Multiple recurrence and convergence for hardy sequences of polynomial growth (2009, result): “We remark that using the corresponding density results (in addition to many other things), Green and Tao ([21]) proved the existence of arbitrarily long arithmetic progressions in the primes, and Tao and Ziegler ([34]) the existence of arbitrarily long polynomial progressions in the primes (this las”
Dense Subsets of Pseudorandom Sets (2008, unknown): “Our functions f are called dual functions in [12], where they are allowed to range over a bounded interval instead of [0] [1], but one can restrict to [0] [1] with no loss of generality after scaling.”