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

TL;DR

This paper establishes a lower bound on the quantity of primes where p-1 is a sum of two squares and p+2 has few prime factors, using advanced sieve techniques.

Contribution

It introduces a novel application of the vector sieve framework to analyze primes with specific additive and multiplicative properties.

Findings

Lower bound for primes with p-1 as sum of two squares and p+2 with bounded prime factors

Application of semi-linear and linear sieve methods in this context

Enhanced understanding of the distribution of such special primes

Abstract

In this paper, we obtain a lower bound for the number of primes $p\leq x$ 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.