Primes in Beatty sequence

Abstract

For a polynomial $g(x)$ of deg $k \geq 2$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $p$ such that $g(p)$ is in non-homogeneous Beatty sequence $\lbrace \lfloor αn+β\rfloor : n=1,2,3, \dots \rbrace$, where $α, β\in \mathbb{R}$ with $α>1$ is irrational and we prove an asymptotic formula for the number of primes $p$ such that $g(p)=\lfloor αn+β\rfloor.$ Next we obtain an asymptotic formula for number of primes $p$ of the form $p=\lfloor αn+β\rfloor$ which also satisfies $p \equiv f \pmod d$ where $f, d$ are integers with $1\leq f < d$ and $(f,d)=1$.