On intervals (kn,(k+1)n) containing a prime for all n>1

TL;DR

This paper investigates which integers k ensure that the interval (kn,(k+1)n) contains a prime for all n>1, identifying all such k up to 50 million and estimating the minimal n for multiple primes.

Contribution

The authors determine all integers k up to 50 million for which (kn,(k+1)n) always contains a prime and provide bounds for the minimal n to find multiple primes in these intervals.

Findings

Identified k=1,2,3,5,9,14 as the only such integers up to 50 million.

Provided upper bounds for the minimal N_k(m) for known k.

Confirmed no other such k exists below 50 million.

Abstract

We study values of k for which the interval (kn,(k+1)n) contains a prime for every n>1. We prove that the list of such integers k includes k=1,2,3,5,9,14, and no others, at least for k<=50,000,000. For every known k of this list, we give a good upper estimate of the smallest N_k(m), such that, if n>=N_k(m), then the interval (kn,(k+1)n) contains at least m primes.