Consider the sequence: $\{p, p^2, \ldots, p^n\}$ where $p \ge 2$ is a prime, and $n \in \mathbb{N}_{>1}$. I’d like to show that:
For all $1 \le i < n$, there exists a$^\dagger$ prime $q$ such that $p^i < q < p^{i+1}$.
For example, the sequence: $\{2, 4, 8, 16, 32\}$ satisfies this property since $2 < 3 < 4$, $4 < 7 < 8$ (or $4 < 5 <8$), $8 < 11 < 16$ (or $13$), $16 < 17 < 32$ (or $23$ etc.).
I am not interested in methods for finding such primes. For now, I am only interested in showing existence of such primes$^\ddagger$. Any pointers?
$^\dagger$ There could be more than one prime $p^i < q < p^{i+1}$, but I’m only interested in showing existence of any one of them.
$^\ddagger$ Context: I'm designing a program where I'm given $p,n$ as input, and available to me a function next_prime()
as well. For the correct functionality of the program, I will call next_prime
($p^i$) and want to make sure that next_prime
($p^i$) will always return a prime $< p^{i+1}$.