  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 $\lbrace
\lfloor \alpha n+\beta\rfloor : n=1,2,3, \dots \rbrace$, 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 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 $1\leq f < d$ and $(f,d)=1$.
