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arXiv · 2404.14765

On the order of magnitude of certain integer sequences

Michael Hellus·Anton Rechenauer·Rolf Waldi
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Abstract

Let $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$ itself: By Harald Helfgott (arXiv:1312.7748 [math.NT]), every odd integer $N$ greater than five can be written as the sum of three prime numbers. There is numerical evidence suggesting that the summands of $N$ always can be chosen between $\frac N6$ and $\frac N2$. This would imply that $u$ is less than $6p$.

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Michael Hellus, Anton Rechenauer, Rolf Waldi. 2024-04-23. On the order of magnitude of certain integer sequences. https://arxiv.org/abs/2404.14765

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