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 $10^7$ and $1.4\cdot 10<sup>9$. In this paper, we extend the verification to all odd integers up to $2<sup>{50}$, thereby improving the previous bound by a factor of more than $8\cdot 10<sup>5$. 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 $&lt;2<sup>{50}$ that require a higher power of two as all smaller ones in their representation.