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

TL;DR

This paper investigates the properties of numbers formed by taking the floor of prime numbers raised to a real power greater than one, establishing the infinitude of such numbers with bounded prime factors and providing explicit bounds.

Contribution

It introduces new results on the distribution of numbers of the form \\lfloor p^c \\rfloor with bounded prime factors for non-integer exponents greater than one.

Findings

Infinitely many \\lfloor p^c \\rfloor have at most R(c) prime factors.

Explicit estimates for R(c) near c=1 and for large c.

Results extend understanding of prime powers and their floor values.

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.