Is it true that $[\pi^n] $ is a prime number for only finitely many integers $n$?

Question

let $[\pi^n] $ be the integer part of $(\pi^n)$, I did mathematica code up to $10^4$ to test primality of $[\pi^n] $ , I have got it could be prime for $n=1,3,4,12$ , Now are there other ? and Is true that $[\pi^n] $ is a prime number for only finitely many integers $n$? ?

Mathematica code

PrimeQ[Table[IntegerPart [Pi^n],{n ,10^4}]]

Answer 1

According to the prime number theorem, the density of primes around $k$ is approximately $1$ in $\log k$. We often approximate this to say $k$ has one chance in $\log k$ of being prime. The expected number of primes is then $$\sum_{n=1}^\infty \frac 1{\log(\lfloor \pi^n \rfloor)}\approx \sum_{n=1}^\infty\frac 1{n \log \pi}$$ which diverges logarithmically, so we would expect infinitely many primes of this form.

Answer 2

According to the OEIS, you overlooked several, starting with

$$\lfloor\pi^{73}\rfloor= 1958577254745770740635072198655932631$$

See also A077547 and A001672.

Note, the OEIS sequence has $10$ primes for $n\le10^4$, which agrees reasonably well with what Ross Millikan's answer would give for an expected number, namely $\sum_{n=1}^{10^4}{1\over n\log\pi}\approx{4\log10\over\log\pi}\approx8$.