Find all values of N and b, (2 ≤ b ≤ 36), such that the base b representations of N^2 and N^3 combined include each digit from 0 to b-1 exactly once.

I know one answer. In base 10, 69^2=4761 and 69^3=328509. This was in my puzzle Pandigital powers. I also know that there cannot be any solutions in a base b of the form 5x+1. If N has x digits, then N^2 has at most 2x digits and N^3 has at most 3x digits. Then, N^2 and N^3 have at most 5x digits together. If N has x+1 digits, then N^2 has at least 2x+1 digits and N^3 has at least 3x+1 digits. Then, N^2 and N^3 have at least 5x+2 digits together. Therefore, N^2 and N^3 cannot have 5x+1 digits, so there are no solutions with b=5x+1.

Posted by Math Man on 2023-11-08 13:50:55