Digital Roots and Powers

When I wake up in the middle of the night, as I often do, I try to put myself back to sleep by doing some sort of mental exercise. Last night I got to playing with digital roots and powers. The digital roots of the squares of numbers with digital roots from 1 to 9, replacing each 3rd number with dashes, since those will all be 9's, are:

1, 4 -- 7, 7 -- 4, 1 a pleasing palindromic result.

Expanding this out a bit, we can construct a table of powers up to 6, as follows (2nd line for square, 3rd line for cube, etc.).

1 2 3 4 5 6 7 8 9
1 4 9 7 7 9 4 1 9
1 8 9 1 8 9 1 8 9
1 7 9 4 4 9 7 1 9
1 5 9 7 2 9 4 8 9
1 1 9 1 1 9 1 1 9

We can see that the 6th power of all non-multiples of 3 have digital root of 1, so the cycle starts all over. We can see also that the powers of both 2 and 5 contain the digits for the famous cyclical number 142857, though the digits are not in the right order. 4 and 7 have a cycle of 3 before repeating, while 8 is a rather boring cycle of 2.

A similar table for the last digit of powers would look like this:

1 2 3 4 5 6 7 8 9
1 4 9 6 5 6 9 4 1
1 8 7 4 5 6 3 2 9
1 6 1 6 5 6 1 6 1
1 2 3 4 5 6 7 8 9

We can see that any number to the 5th power will end in the same digit which the original number ended in, so we only have to go up to 4 powers to have a complete table ad infinitum, unlike with digital roots where we needed to go to the 6th power.

We have seen that the digital root of a power of 2 is always 1, 2, 4, 5, 7, or 8. It so happens that these are also the digital roots of all the prime numbers (except for 3). (This is per Wikipedia, I don't pretend to understand why this is so.) More Wikipedia goodies: digital root of a perfect number will always be 1; digital root of a triangular number will always be 1, 3, 6, or 9.