Is there a fixed integer $n$ for which ${\pi}^{n}$ is prime number?

Question

I would like to know the relationship between $\pi$ and prime numbers distribution ,then I would like to ask if there is a fixed integer for which ${\pi}^{n}$ can be prime or how do i disproof that is can not be prime for any arbitrary integer $n $ ?

Thank you for any help

Answer 1

$\pi$ is transendental which means it is not the root to any polynomial with rational coefficients. That means $x^n - z = 0$ will never have $\pi$ as a root. So $\pi^n =z \in \mathbb Z$ will never happen. So $\pi^n$ is never a prime because it is never an integer.