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

Likun Xie

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 $2^{k(N)} \leq N^θ$ for some $θ> 0$ 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.

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

Abstract

We prove lower bounds for the number of primes such that is divisible by and has at most odd prime factors (), assuming for some depending on . 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.

Paper Structure

This paper contains 4 sections, 20 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. p - b divisible exactly by 2^k(N) and at most two odd prime divisors
  3. Bounds for the Sieving Functions
  4. Application of Richert's Weight: $p - b$ Divisible by Larger Powers of $2$

Key Result

Theorem 1.1

Let $b > 1$ be an odd integer, and define $\pi_b(x)$ to be the number of primes of the form $2^n + b$ with $n \leq x$. Then provided both the Extended Riemann Hypothesis and Hypothesis A hooley hold.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2: prime_power
  • Lemma 1.1: prime_power
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.1
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Lemma 2.1
  • ...and 21 more