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/( | 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/(|m√2| - |m√1.5|)^m there exist a prime number r and a positive integer a such that n < r < 3n/2 < a^m < 2n.