How often are $ \lfloor {n^{\alpha}} \rfloor $ and $ \lfloor {n^{\beta}} \rfloor $ simultaneously primes?

TL;DR

This paper proves that for certain exponents, there are infinitely many integers where all the floored powers are prime, using a new equidistribution theorem across multiple arithmetic progressions.

Contribution

It introduces a novel simultaneous equidistribution theorem for floored powers and demonstrates its application to prime occurrence in multiple sequences.

Findings

Infinitely many integers n with all floored powers prime for certain exponents.

Established a simultaneous equidistribution theorem for floored powers.

Extended prime distribution results to multiple sequences simultaneously.

Abstract

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<\alpha_1 < \alpha_2 < \cdots< \alpha_k < 1$ satisfying a certain condition, we show that there are infinitely many positive integers $n$ for which all of $ \lfloor{n^{\alpha_1}}\rfloor, \lfloor{n^{\alpha_2}}\rfloor,\ldots, \lfloor{n^{\alpha_k}}\rfloor $ are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for $ \lfloor{n^{\alpha_i}}\rfloor $ across $k$-many arithmetic progressions.