On the Sum of Squarefree Integers and a Power of Two

TL;DR

This paper verifies Erdos's conjecture for all odd numbers up to 2^50 using a GPU-accelerated algorithm, significantly extending previous computational bounds and providing new numerical insights.

Contribution

The paper extends the computational verification of Erdos's conjecture to all odd integers up to 2^50 using a highly parallelized GPU algorithm, surpassing previous bounds by over 8×10^5.

Findings

Verification of the conjecture up to 2^50

Identification of the smallest odd numbers requiring larger powers of two

Development of a GPU-based highly parallelized algorithm

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.