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.

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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).

I can tell you why the digital root of all the prime numbers (except for 3) is always 1, 2, 4, 5, 7, or 8. It is because all the prime numbers can be written as "6k+1" or "6k-1" (where "k" is optional). And the digital root of a power of 5 or 7 is always 1, 2, 4, 5, 7, or 8.

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