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

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 $\lfloor n^{{\alpha }_1}\rfloor ,\lfloor n^{{\alpha }_2}\rfloor ,\dots ,\lfloor n^{{\alpha }_k}\rfloor$ are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for $\lfloor n^{{\alpha }_i}\rfloor$ across $k$-many arithmetic progressions.

11 pages