Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the upper and lower bound orders of almost prime triples

Runbo Li

Abstract

A Hardy-Littlewood triple is a 3-tuple of integers with the form $(n, n+2, n+6)$. In this paper, we study Hardy-Littlewood triples of the form $(p, P_{a}, P_{b})$ and improve the upper and lower bound orders of it, where $p$ is a prime and $P_{r}$ has at most $r$ prime factors. Our new results generalize and improve the previous results.

On the upper and lower bound orders of almost prime triples

Abstract

A Hardy-Littlewood triple is a 3-tuple of integers with the form . In this paper, we study Hardy-Littlewood triples of the form and improve the upper and lower bound orders of it, where is a prime and has at most prime factors. Our new results generalize and improve the previous results.
Paper Structure (9 sections, 11 theorems, 83 equations)

This paper contains 9 sections, 11 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Weighted sieve method
  4. Proof of Theorem 1.1
  5. Evaluation of $S_{1}, S_{2}, S_{3}$
  6. Evaluation of $S_{4}, S_{5}, S_{6}$
  7. Evaluation of $S_{7}$
  8. Proof of theorem 1.1
  9. An upper bound result

Key Result

Theorem 1.1

For every integer $a \geqslant 2$ and $b \geqslant 14$, we have where $\pi_{1,a,b}(x)$ and $D_{1,a,b}(N)$ are defined above.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Conjecture 2.4
  • Lemma 2.5
  • ...and 3 more