# Primes in Beatty sequence

## TL;DR

This paper investigates primes within Beatty sequences generated by irrational slopes, providing bounds on the least such prime, asymptotic counts for primes of specific forms, and their distribution properties.

## Contribution

It establishes new upper bounds and asymptotic formulas for primes in Beatty sequences, including primes of polynomial forms and with specified congruence conditions.

## Key findings

- Bound on the least prime in Beatty sequence where polynomial takes prime values
- Asymptotic formula for the count of primes of the form loorlpha n+eta
- Distribution results for primes in Beatty sequences with congruence restrictions

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