# Proof that. for any two natural numbers $n$, $m$, $n^m$ is not an $n$-ary pernicious number.

$a_{10}$ is defined to be an $n$-ary pernicious number when the digit sum of $a_n$ is prime in base $10$.

How can I prove that, for any two natural numbers $n$, $m$, $n^m$ is not an $n$-ary pernicious number? I thought about using mathematical induction, but I have no idea how to go about doing this.

(I cannot even prove the trivial case $n=2$, much less any other case!)

The form of all such numbers in base $n$ is $1000...000$, or $1$ followed by $m-1$ zeroes.
Hence the sum of all digits in each number of form $n^m$, base $n$, is $1$, which is not prime.