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On the Sum of Squarefree Integers and a Power of Two

Christian Hercher

Abstract

Erdos conjectured that every odd number greater than one can be expressed as the sum of a squarefree number and a power of two. Subsequently, Odlyzko and McCranie provided numerical verification of this conjecture up to $10^7$ and $1.4\cdot 10^9$. In this paper, we extend the verification to all odd integers up to $2^{50}$, thereby improving the previous bound by a factor of more than $8\cdot 10^5$. Our approach employs a highly parallelized algorithm implemented on a GPU, which significantly accelerates the process. We provide details of the algorithm and present novel heuristic computations and numerical findings, including the smallest odd numbers $<2^{50}$ that require a higher power of two as all smaller ones in their representation.

On the Sum of Squarefree Integers and a Power of Two

Abstract

Erdos conjectured that every odd number greater than one can be expressed as the sum of a squarefree number and a power of two. Subsequently, Odlyzko and McCranie provided numerical verification of this conjecture up to and . In this paper, we extend the verification to all odd integers up to , thereby improving the previous bound by a factor of more than . Our approach employs a highly parallelized algorithm implemented on a GPU, which significantly accelerates the process. We provide details of the algorithm and present novel heuristic computations and numerical findings, including the smallest odd numbers that require a higher power of two as all smaller ones in their representation.

Paper Structure

This paper contains 4 sections, 3 theorems, 11 equations, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Heuristic arguments
  3. Algorithms
  4. Results

Key Result

Lemma 2

Let $0<x\leq \frac{1}{9}$ be a real number and $2\leq k \leq 20$ be an integer with $kx \leq \frac{4}{5}$. Then Here and in the following $\log$ denotes the natural logarithm.

Theorems & Definitions (7)

  • Conjecture 1: Erdős
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 4
  • Theorem 5