Clusters of primes with square-free translates

TL;DR

This paper proves the existence of long clusters of primes within bounded intervals that satisfy specific squarefree translate conditions and can be constrained to various special sets or intervals, advancing understanding of prime distribution.

Contribution

It introduces new methods to find prime clusters with squarefree translates in bounded intervals, including inhomogeneous Beatty sequences and fractional power residue conditions.

Findings

Existence of long prime clusters with squarefree translates in bounded intervals.

Prime clusters can be constrained to inhomogeneous Beatty sequences.

Prime clusters can satisfy fractional power residue conditions.

Abstract

Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal{R}$. Moreover, we can enforce that the primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N, N+N^{\frac{7}{12}+\epsilon}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c \in (\frac{8}{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+\epsilon}$.