Almost primes and primes that are sums of two squares plus 1

Abstract

In this paper, we obtain a lower bound for the number of primes $p\le x$ such that $p-1$ is a sum of two squares and $p+2$ has a bounded number of prime factors. The proof uses the vector sieve framework, involving a semi-linear sieve and a linear sieve.

Final version, published in Acta Arithmetica