Primes of the form $[n^c]$ with square-free $n$

TL;DR

This paper proves that for certain exponents between 1 and approximately 1.154, there are infinitely many primes of the form floor(n^c) with n being square-free, expanding understanding of prime distribution in special sequences.

Contribution

It establishes the existence of infinitely many primes of the form floor(n^c) with square-free n for a specific range of c, which was previously unknown.

Findings

Infinitely many primes of the form floor(n^c) exist for 1 < c < 3849/3334.

The result applies specifically to n being square-free.

The proof extends the understanding of prime distribution in fractional power sequences.

Abstract

Let $[\, \cdot\,]$ be the floor function. In this paper we show that when $1<c<\frac{3849}{3334}$, then there exist infinitely many prime numbers of the form $[n^c]$, where $n$ is square-free.