A conjecture: every power of $16$ greater then $16^4$ has at least one digit $1, 2, 4$, or $8$ when written as base $10$? Is there a proof for why the following is true? Does every power of $16$ greater then $16^4$ have a digit $1, 2, 4$ or $8$ when you write it as base $10$?
Maple yields that ${\left(16^4\right)}^{20}$ is 

2135987035920910082395021706169552114602704522356652769947041607822219725780640550022962086936576


whose digits contain more than one instance of each of $1,2,4$ and $8$, so the answer to your question would most definitely be no.
Maple yields that ${\left(16^4\right)}^{20}$ is 

2135987035920910082395021706169552114602704522356652769947041607822219725780640550022962086936576


whose digits contain more than one instance of each of $1,2,4$ and $8$, so the answer to your question would most definitely be no.