Revista Matemática Iberoamericana


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Volume 33, Issue 3, 2017, pp. 809–829
DOI: 10.4171/RMI/956

Published online: 2017-10-02

Clusters of primes with square-free translates

Roger C. Baker[1] and Paul Pollack[2]

(1) Brigham Young University, Provo, USA
(2) University of Georgia, Athens, USA

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 $[N, N+N^{{7}/{12}+\epsilon}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c \in ({8}/{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+\epsilon}$.

Keywords: Maynard–Tao method, primes with square-free translates, mixed exponential sums

Baker Roger, Pollack Paul: Clusters of primes with square-free translates. Rev. Mat. Iberoamericana 33 (2017), 809-829. doi: 10.4171/RMI/956