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arXiv · 2411.01964

On the Sum of Squarefree Integers and a Power of Two

Christian Hercher
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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.

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Christian Hercher. 2024-11-04. On the Sum of Squarefree Integers and a Power of Two. https://arxiv.org/abs/2411.01964

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