The Green-Tao theorem for primes of the form $x^2+y^2+1$

TL;DR

This paper proves that there are infinitely many primes of the form x^2 + y^2 + 1 that form arbitrarily long arithmetic progressions, extending understanding of prime distributions in special quadratic forms.

Contribution

It establishes the existence of arbitrarily long arithmetic progressions within primes of the specific form x^2 + y^2 + 1, a significant extension of Green-Tao type results.

Findings

Primes of the form x^2 + y^2 + 1 contain arbitrarily long arithmetic progressions.

The result generalizes Green-Tao theorem to a new class of primes.

Provides new insights into the distribution of primes in quadratic forms.

Abstract

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