Clusters of primes with square-free translates

2015/05/11 by Baker, Roger C., Pollack, Paul
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1505.02744

Abstract

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

Related

• Dense clusters of primes in subsets
• Short intervals containing a prescribed number of primes
• Arithmetic progressions of primes in short intervals beyond the 17/30 barrier
• Squarefree smooth numbers and Euclidean prime generators
• Bounded prime gaps in short intervals