The Gaussian primes contain arbitrarily shaped constellations

@article{Tao2005TheGP,
  title={The Gaussian primes contain arbitrarily shaped constellations},
  author={Terence Tao},
  journal={Journal d’Analyse Math{\'e}matique},
  year={2005},
  volume={99},
  pages={109-176}
}
  • T. Tao
  • Published 20 January 2005
  • Mathematics
  • Journal d’Analyse Mathématique
We show that the Gaussian primesP[i] ⊆ ℤ[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, we show that given any distinct Gaussian integersv0,…,vk−1, there are infinitely many sets {a+rv0,…,rvk−1}, witha ∈ℤ[i] andr ∈ℤ{0}, all of whose elements are Gaussian primes.The proof is modeled on that in [9] and requires three ingredients. The first is a hypergraph removal lemma of Gowers and Rödl-Skokan or, more precisely, a slight strenghthening of this… 
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