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 $\{\lfloor \alpha n+\beta \rfloor :n=1,2,3,\dots \}$, 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 $p\equiv f(\mathrm{mod}d)$ where $f,d$ are integers with $1\le f<d$ and $(f,d)=1$.

12 pages, 0 figures

References in corpus (1)

• The Least Prime Number in a Beatty Sequence