On the Interval [n,2n]: Primes, Composites and Perfect Powers

In this paper we show that for every positive integer $n$ there exists a prime number in the interval $[n,9(n+3)/8]$. Based on this result, we prove that if $a$ is an integer greater than 1, then for every integer $n>14.4a$ there are at least four prime numbers $p$, $q$, $r$, and $s$ such that $n<ap<3n/2<aq<2n$ and $n<r<3n/2<s<2n$. Moreover, we also prove that if $m$ is a positive integer, then for every positive integer $n>14.4/(|\sqrt[m]{1.5}|-1)^m$ there exist a positive integer $a$ and a prime number $s$ such that $n<a^m<3n/2<s<2n$, as well as the fact that for every positive integer $n>14.4/(|\sqrt[m]{2}|-|\sqrt[m]{1.5}|)^m$ there exist a prime number $r$ and a positive integer $a$ such that $n<r<3n/2<a^m<2n$.