Primes $p$ such that $p-b$ Has a Large Power Factor and Few Other Prime Divisors

TL;DR

This paper establishes lower bounds on the quantity of primes p where p-b has a large power of 2 dividing it and a limited number of odd prime factors, using advanced sieve methods and prime distribution results.

Contribution

It introduces new bounds for primes with specific divisibility properties of p-b, extending sieve techniques and prime distribution theories.

Findings

Lower bounds for primes with p-b divisible by large powers of 2

Primes p where p-b has at most k odd prime factors

Application of weighted sieves and Elliott's results

Abstract

We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^\theta$ for some $\theta > 0$ depending on $k$. The proof uses a variant of Chen's method, weighted sieves, and Elliott's results on primes in arithmetic progressions with large power-factor moduli.