Powers containing every digit equally often

Question

There are several nontrivial powers containing every digit equally often, for example

$32043^2$ $2158479^3$ $69636^4$ $643905^5$ $3470187^6$

A necessary condition for a power with the desired property is that the base is divisible by $3$ because the power must be a multiple of $9$.

My questions :

• Are there powers of $3$,$6$ and $9$ with the desired property ?
• Are there infinite many nontrivial powers with the desired property ?
• I did not find a $10th$ power with the desired property yet. Is there one ?

Answer 1

Here is a $\frac{1}{3}$th of an answer. There is an 10th power with the property. Here is a list of the smallest solutions for a given power:

Edit to keep the post small, see Jeppes comment for a list up to 24:

• 10: 81785058

Answer 2

This is not a proof, but a heuristic argument.

Of the $10^{10m} - 10^{10m-1}$ positive integers with $10m$ digits, $\dfrac{9}{10} \dfrac{(10m)!}{m!^{10}} \approx \dfrac{9}{32 \sqrt{5}} m^{-9/2} 10^{10m}$ have all digits equally often. Now the number of $k$'th powers with $10m$ digits is approximately $(1 - 10^{-1/k}) 10^{10m/k}$, so we should expect the number of $k$'th powers with $10m$ digits and the desired property to be on the order of $ m^{-9/2} 10^{10m/k}$. In particular, for any $k$ there should be infinitely many $k$'th powers with the property.