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arXiv · 2609.20505

Computational results on sums of a prime with squares or cubes

Kenny Applegate·Kyle Pratt
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Abstract

One conjectures that all integers greater than 14 are the sum of an odd prime and two positive squares. We prove that all integers $n$ with $14<n<10^{26}$ can be represented in this manner. Along the way, we investigate sums of a prime and one square and then ``bootstrap'' to get our results on sums of a prime and two squares. We also show that all integers between $308$ and $10^{16}$ can be represented as the sum of an odd prime and two positive cubes. Our methods require the determination of the integral points on various elliptic curves.

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Kenny Applegate, Kyle Pratt. 2026-09-17. Computational results on sums of a prime with squares or cubes. https://arxiv.org/abs/2609.20505

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