I've seen on wikipedia this formula:
$$\pi_k(p^n) = p^{n-1}\pi_k(p) $$
It says that the formula holds unless p is k-Wall-Sun-Sun prime, or k-Fibonacci-Wieferich prime, that is, $p^2$ divides $F_k(p-1)$ or $F_k(p+1)$.
I don't know if the statement was proven or not. I've search on the internet but could not found any proof for this.
I'm interested in the case when $p = 2$, that is if $\pi_k(2^n) = 2^{n-1}\pi_k(2) $ was proven or it was just tested with computers. I know that the formula true, but I need the proof of the formula if it exists, because I want to use the formula in my work.
Best Answer
It's not a conjecture and the formula was proven. If p is not k-Wall-Sun-Sun prime then the formula holds. There's no known prime number which is a k-Wall-Sun-Sun. Prime numbers till $10^{14}$ were tested with computers. Thus, for p = 2 the formula was proven.