Almost primes and primes that are sums of two squares plus one
Kunjakanan Nath, Likun Xie
TL;DR
The paper proves a lower bound for primes $p\le x$ with $p = m^2+n^2+1$ and $\Omega(p+2) \le 11$, showing $\#\{p\le x: p\text{ prime}, p=m^2+n^2+1, \Omega(p+2)\le 11\} \gg \frac{x}{(\log x)^{5/2}}$ via a vector sieve that combines a semi-linear sieve ($\kappa=1/2$) and a linear sieve ($\kappa=1$). It introduces a weighted component to optimize the $p+2$ factorization and uses a switching principle to handle cross-sieve sums, with a fundamental vector-sieve lemma controlling two beta sieves. The work situates itself in the context of prime k-tuple and almost-prime correlations, extending classical results on primes in sparse polynomial families. Under stronger distributional hypotheses, one can push the bound to smaller $\Omega(p+2)$, highlighting the potential of vector-sieve methods in probing rare prime configurations. Overall, the paper advances the understanding of primes arising from binary quadratic forms and their near-prime companions through intricate sieve technology and careful parameter optimization.
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.