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On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors

Ethan S. Lee, Rowan O'Clarey

Abstract

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove new results on square-free numbers co-prime to any integer with up to two prime factors, which make the expected asymptotic results explicit.

On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors

Abstract

Every natural number greater than can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove new results on square-free numbers co-prime to any integer with up to two prime factors, which make the expected asymptotic results explicit.

Paper Structure

This paper contains 8 sections, 14 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Computations
  3. Bounds for the number of representations
  4. Prime number theorems
  5. Bounds for $T(n)$ and $R(n)$
  6. Bounds for $T_q(n)$ and $R_q(n)$ when $q>3$
  7. Bounds for $T_3(n)$ and $R_3(n)$
  8. Proof of main result

Key Result

Theorem 1.1

Fix any integer $k > 1$ with at most two prime factors.

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 19 more