The primes contain arbitrarily long polynomial progressions

Terence Tao Department of Mathematics, University of California, Los Angeles Tamar Ziegler Department of Mathematics, Technion – Israel Institute of Technology

TBD mathscidoc:1701.331999

Acta Mathematica, 201, (2), 213-305, 2006.10
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.
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@inproceedings{terence2006the,
  title={The primes contain arbitrarily long polynomial progressions},
  author={Terence Tao, and Tamar Ziegler},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203352583751708},
  booktitle={Acta Mathematica},
  volume={201},
  number={2},
  pages={213-305},
  year={2006},
}
Terence Tao, and Tamar Ziegler. The primes contain arbitrarily long polynomial progressions. 2006. Vol. 201. In Acta Mathematica. pp.213-305. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203352583751708.
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