The Green–Tao theorem for primes of the form \(x^2+y^2+1\)

  • Yu-Chen Sun
  • Hao Pan


We prove that the primes of the form \(x^2+y^2+1\) contain arbitrarily long non-trivial arithmetic progressions.


Prime Arithmetic progression Pseudorandom measure 

Mathematics Subject Classification

Primary 11P32 Secondary 11B25 11B30 11N36 



We are grateful to the anonymous referee for his/her very helpful comments. We also thank Professor Henryk Iwaniec for his helpful explanation on Theorem 1 of [7]. The work is supported by National Natural Science Foundation of China (Grant No. 11671197). The second author is the corresponding author.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Medical SchoolNanjing UniversityNanjingPeople’s Republic of China
  2. 2.School of Applied MathematicsNanjing University of Finance and EconomicsNanjingPeople’s Republic of China

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