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Infinite partial sumsets in the primes

Terence Tao, Tamar Ziegler

Abstract

We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$.

Infinite partial sumsets in the primes

Abstract

We show that there exist infinite sets and of natural numbers such that is prime whenever .
Paper Structure (7 sections, 10 theorems, 53 equations)

This paper contains 7 sections, 10 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Acknowledgements
  4. Proof of Theorem \ref{['main1']}
  5. An application of Bergelson's intersectivity lemma
  6. The Maynard sieve
  7. Consequences for the orbit closure of the primes

Key Result

Theorem 1.3

Assume Conjecture qhl. Then there exists an infinite set $B$ of primes such that $b+b'+1$ is prime for every distinct $b,b' \in B$.

Theorems & Definitions (22)

  • Definition 1.1: Admissible and prime-producing tuples
  • Conjecture 1.2: Dickson--Hardy--Littlewood conjecture
  • Theorem 1.3: Infinite restricted sumsets in the shifted primes
  • Remark 1.4
  • Theorem 1.5: Ascending chain of prime-producing tuples
  • Corollary 1.6: Primes contain half of an infinite sumset
  • proof
  • Proposition 2.1: Extending a good tuple
  • proof
  • Remark 2.2
  • ...and 12 more