Does there exist a prime $p \geq 7$ such that the order of $4$ in the multiplicative group of units in $\mathbb{Z}/p^n$ is odd for every positive integer $n$?
It would be nice if $7$ was already an example. I computed the order of $4$ modulo $7^n$ for $n=1,2,\ldots,12$ and it came out odd, but that's not particularly compelling evidence I guess.