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arXiv · 1508.03281

Some arithmetic properties of numbers of the form $\lfloor p^c\rfloor$

William D. Banks·Victor Z. Guo·Igor E. Shparlinski
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Abstract

Let $${\mathbb P}^c=(\lfloor p^c\rfloor)_{p\in{\mathbb P}} \qquad (c>1,\ c\not\in {\mathbb N}), $$ where ${\mathbb P}$ is the set of prime numbers, and $\lfloor\cdot\rfloor$ is the floor function. We show that for every such $c$ there are infinitely many members of ${\mathbb P}^c$ having at most $R(c)$ prime factors, giving explicit estimates for $R(c)$ when $c$ is near one and also when $c$ is large.

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William D. Banks, Victor Z. Guo, Igor E. Shparlinski. 2015-08-13. Some arithmetic properties of numbers of the form $\lfloor p^c\rfloor$. https://arxiv.org/abs/1508.03281

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