Arithmetic progressions of three prime numbers with two primes of the form $\mathbf{p}={\mathbf{x}}^2+{\mathbf{y}}^2+1$

S. I. Dimitrov

In the present paper we prove that there exist infinitely many arithmetic progressions of three different primes $p_1,p_2,p_3=2p_2-p_1$ such that $p_1=x_1^2+y_1^2+1$, $p_2=x_2^2…