How often are $ \lfloor {n^α} \rfloor $ and $ \lfloor {n^β} \rfloor $ simultaneously primes?

TL;DR

This study investigates the simultaneous primality of Beatty-type sequences $\lfloor n^{\alpha_i}\rfloor$ by proving a high-dimensional equidistribution result for these floor values across arithmetic progressions. The core tool is a multivariable uniform distribution bound, enabling an asymptotic for products of von Mangoldt values and hence infinitely many $n$ with all $\lfloor n^{\alpha_i}\rfloor$ prime under a technical exponent constraint. It also derives corollaries (notably for $k=2$) and extends the method to squares, divisors, and sum-of-divisors functions, yielding corresponding asymptotics. The work highlights both the potential of simultaneous equidistribution in Beatty-type sequences and the current limitations tied to exponent bounds, pointing to avenues for sharpening error terms and broadening the prime‑producing regime.

Abstract

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<α_1 < α_2 < \cdots< α_k < 1$ satisfying a certain condition, we show that there are infinitely many positive integers $n$ for which all of $ \lfloor{n^{α_1}}\rfloor, \lfloor{n^{α_2}}\rfloor,\ldots, \lfloor{n^{α_k}}\rfloor $ are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for $ \lfloor{n^{α_i}}\rfloor $ across $k$-many arithmetic progressions.

Theorems & Definitions (14)

• Theorem 1.1
• Corollary 1
• Theorem 1.2
• Corollary 2
• Proposition 2.1
• Lemma 2.2: Effective Poisson summation formula
• Lemma 2.3
• proof : Proof of Proposition \ref{['unif-distribution']}
• Lemma 2.4
• proof