Consecutive primes and Beatty sequences

TL;DR

This paper investigates the distribution of consecutive prime pairs within specific Beatty sequences, providing an asymptotic count under Hardy-Littlewood conjectures, linking prime distribution to irrational number sequences.

Contribution

It establishes an asymptotic formula for counting consecutive primes in Beatty sequences assuming Hardy-Littlewood conjectures, connecting prime gaps with irrational sequences.

Findings

Asymptotic count of prime pairs in Beatty sequences derived

Conditional on Hardy-Littlewood conjectures, the distribution aligns with predicted densities

Provides a quantitative link between prime distribution and irrational number sequences

Abstract

Fix irrational numbers $\alpha,\hat\alpha>1$ of finite type and real numbers $\beta,\hat\beta\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloor\alpha m+\beta\rfloor)_{m\ge 1}\quad\text{and}\quad\hat B:=(\lfloor\hat\alpha m+\hat\beta\rfloor)_{m\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\sharp)$ of consecutive primes for which $p\in B$ and $p^\sharp\in\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \big|\{p\le x:p\in B\text{ and }p^\sharp\in\hat B\}\big|=(\alpha\hat\alpha)^{-1}\pi(x)+O\big(x(\log x)^{-3/2+\epsilon}\big), $$ where $\pi(x)$ is the prime counting function.