BG-MO 1988 Problem 2

Let $n$ and $k$ be natural numbers and $p$ a prime number. Prove that if $k$ is the exact exponent of $p$ in $2^{2^n}+1$ (i.e. $p^k$ divides $2^{2^n}+1$ , but $p^{k+1}$ does not), then $k$ is also the exact exponent of $p$ in $2^{p-1}-1$ .