If you only knew the magnificence of the 3, 6 and 9, then you would have a key to the universe. — Nikola Tesla (attributed)
There are a number of videos on youtube showing some interesting arithmetic patterns regarding digital roots (call it $T(•)$). For instance, $\forall n \in \mathbb{N}$:
- $T(2^n)=x_{mod(n,6)}, x=(1,2,4,8,7,5)$
- $T(5^n)=x_{mod(n,6)}, x=(1,5,7,8,4,2)$
- $T(4^n)=x_{mod(n,3)}, x=(1,4,7)$
- $T(7^n)=x_{mod(n,3)}, x=(1,7,4)$
- $T(8^n)=x_{mod(n,2)}, x=(1,8)$
And $\forall n>1$:
- $T(3^n)=T(6^n)=T(9^n)=9$
Also, $\forall n \in \mathbb{N}$:
- $T(3(n+1))=x_{mod(n,3)}, x=(3,6,9)$
These sequences are represented in the following diagram by Randy Powell:
Another version was drawn by Math teacher Joey Grether (who attributed it to Tesla as a hoax, after unsuccessfully trying to divulge it):
So the question is: is there a mathematical structure behind all this, or is it just a simple number game?