[Guest]

Each A of B is a positive integer with A ≤ B, and N is a positive integer ≥ 3 such that the base ten representations of A^N and B^N will together contain each of the digits from 1 to 9 exactly once.

Determine the minimum value of A^N and the maximum value of B^N.

[superprismatic]

The minimum value of A^N is 8;

The maximum value of B^N is 32461759.

This occurs when N=3, A=2, and B=319.

[superprismatic]

If, however, we take 'A^N and B^N will together

contain each of the digits from 1 to 9 exactly once' to implicitly

allow as many imbedded 0 digits as we like....

B^N could be 961504803 (with A^N=27) using

N=3, A=3, and B=987. But, I'm not sure it's maximal. In any

case, A^N still has its minimum at 8 (with either

B^N=24137569 [N=3, A=2, B=289] or B^N=32461759

[N=3, A=2, B=319])

Edited January 9, 2010 by superprismatic