N is a m-th power of an integer.
It consists of n distinct digits, averaging n.

Find n,m and N, using your head + pen and paper only.

Check, whether there exist additional solutions (computer allowed).

Clearly, n>1; m>1 - to exclude trivial cases.

Squares:
3 digits count: 5
4 digits count: 6 {41,58,59,68,85,95}
5 digits count: 19 {113,...}
6 digits count: 38 {387,...}
7 digits count: 14 {1667,...}

Cubes
6^3 = 216
66^3, 96^3, 99^3 all have 6 digits averaging 6

Others
2^16 = 4^8 = 16^4 (=256^2) has 5 digits averaging 5

3^5 = 243
15^5 = 759375
6^7 = 279936

This list may be exhaustive (my counting may also be off.)  I didn't look at 8 digit squares but I don't expect there are any.

Posted by Jer on 2014-01-27 13:39:16