Primes in Beatty sequence

Abstract

For a polynomial g(x) of $\deg k \ge 2$ 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 $\lbrace \lfloor \alpha n+\beta \rfloor : n=1,2,3, \dots \rbrace$, where $\alpha , \beta \in {\mathbb {R}}$ with $\alpha >1$ is irrational and we prove an asymptotic formula for the number of primes p such that $g(p)=\lfloor \alpha n+\beta \rfloor .$ Next, we obtain an asymptotic formula for the number of primes p of the form $p=\lfloor \alpha n+\beta \rfloor$ which also satisfies $p \equiv f \pmod d$, where f, d are integers with $1\le f < d$ and $(f,d)=1$.