# 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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