Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Consecutive primes and IP sets

William D. Banks

Abstract

For an infinite set M of natural numbers, let FS(M) be the set of all nonzero finite sums of distinct numbers in M. An IP set is any set of the form FS(M). Let p_n denote the n-th prime number for each $n \ge 1$. A de Polignac number is any number m such that $p_{n+1}-p_n=m$ for infinitely many n. In this note, we show that every IP set of even natural numbers contains infinitely many de Polignac numbers.

Consecutive primes and IP sets

Abstract

For an infinite set M of natural numbers, let FS(M) be the set of all nonzero finite sums of distinct numbers in M. An IP set is any set of the form FS(M). Let p_n denote the n-th prime number for each . A de Polignac number is any number m such that for infinitely many n. In this note, we show that every IP set of even natural numbers contains infinitely many de Polignac numbers.
Paper Structure (7 sections, 9 theorems, 12 equations)

This paper contains 7 sections, 9 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Background Material
  3. de Polignac numbers
  4. The Maynard-Tao theorem
  5. Ramsey's theorem
  6. Statement of the main theorem
  7. Proof of the main theorem

Key Result

Theorem 1

For every IP set $\mathcal{A} \subset 2\mathbb N$, there exist infinitely many $a\in\mathcal{A}$ with the property that $p_{n+1}-p_n=a$ for infinitely many $n$.

Theorems & Definitions (11)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Theorem 5: Maynard--Tao
  • Theorem 6: Banks--Freiberg--Turnage-Butterbaugh
  • Theorem 7: Ramsey
  • Theorem 8
  • Lemma 9
  • proof
  • ...and 1 more