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A Density Theorem for Higher Order Sums of Prime Numbers

Michael T. Lacey, Hamed Mousavi, Yaghoub Rahimi, Manasa N. Vempati

Abstract

Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for $k \geq 4$ distinct subsets of primes. This extends the work of H.~Li, H.~Pan, as well as X.~Shao on sums of three primes, and A.~Alsteri and X.~Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas.

A Density Theorem for Higher Order Sums of Prime Numbers

Abstract

Let be a subset of the primes of lower density strictly larger than . Then, every sufficiently large even integer is a sum of four primes from the set . We establish similar results for -summands, with , and for distinct subsets of primes. This extends the work of H.~Li, H.~Pan, as well as X.~Shao on sums of three primes, and A.~Alsteri and X.~Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas.

Paper Structure

This paper contains 7 sections, 11 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Key Arithmetic Lemmas
  3. Proof of Lemma \ref{['t:Zqk']}
  4. Combinatorial Lemmas
  5. A Single Set of Primes
  6. Four and More Sets of Primes
  7. Proof of the theorems

Key Result

Theorem 1.2

Let $k\geq 4$. Let $P \subseteq \mathbb{P}$ be a subset of primes with the density $d(P) > \frac{1}{2}$. Then there is an integer $N_{P,k}$ so that for every integer $n > N_{P,k}$ congruent to $k$ mod $2$, there are $p_i \in P$, $1\leq i \leq k$ such that $n = \sum_{i=1}^k p_i$.

Theorems & Definitions (15)

  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.4: Cauchy-Davenport Theorem
  • Lemma 3.1
  • Remark 3.9
  • Lemma 3.10
  • proof
  • Lemma 3.19
  • ...and 5 more