MHB Recreational Number Theory, Unsolved Problem

[Tamas]

Find a perfect power k^m > 1 where k, m, k^m do not contain 2 in their decimal digits, nor do share any decimal digit, no matter if k^m might possibly be expressed in more than one way for some value, e.g. 8^2 = 4^3. I do not know if such an integer exists at all, or how many and how large they are if they do. What did I do to try finding a solution to this problem? I cannot compute or program, so I tried an online big integer calculator with manual input and checking. This was, though methodical, but slow. I got to very large numbers without success, and the more digits appeared, the less likelihood remained for finding a match. Since I am not a mathematician, let alone a number theorist, I cannot prove or disprove the existence of such integer. Finding one can be a proof, but it is beyond my capabilities. Still, this interesting problem fascinates me and I hope others will like it too.

 

[Tamas]

The more interesting it is, because powers exist with all the other individual decimal digits d missing from the otherwise also not decimal digit sharing k, m, and k^m.
So, d = 2 seems to be elusive, or, is indeed the exception?
Easily found examples for each d not equal 2 as follows:
For d = 0 -> 2^3 = 8; for d = 1 -> 3^2 = 9; for d = 3 -> 67^2 = 4489; for d = 4 -> 33^2 = 1089; for d = 5 -> 2^4 = 4^2 = 16; for d = 6 -> 7^2 = 49;
for d = 7 -> 44^2 = 1936; for d = 8 -> 34^2 = 1156; and for d = 9 -> 38^2 = 1444.
I believe a brute force search may bring up perhaps an example for d = 2, or an insightful proof is found for its impossibility and therefore non-existence.
Without these, we don't know.