Resumen de Clusters of primes with square-free translates

Roger C. Baker, Paul Pollack

• Let 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∈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+N7/12+ϵ], (2) p belongs to a given inhomogeneous Beatty sequence, (3) with c∈(8/9,1) fixed, pc lies in a prescribed interval mod 1 of length p−1+c+ϵ.