I am still painfully obsessed with digital roots. No luck in constructing a proof for the persistance of the patterns, though it may not be necessary. Digital Roots contain additive and multiplicitive properties, so it is highly possible that there is just some natural fallout occuring that produces the patterns. I wish I knew proofs better, but alas I do not.
A new curiosity that has crept in though is that the Digital Roots can be used to find a likely prime, and to potentially prove a prime. Again I can't find any pre-existing information in this area, but I have some very cute programs that demonstrate it well enough. In fact, David Barksdale (aka amatus) has come up with a proof for one of my two digital root prime finders that shows it to be the equivilant of Fermat's Little Therom, and thus it is subject to the same shortfalls i.e. Carmichael Numbers are also found. So while it is nothing knew, it is neat to think that it evolved from something I came up with while trying to sleep at night:
b-1 is prime if for 1 .. b, DRb(n ** 1) == DRb(n ** (b-1)))
The second algorithm I have is more curious as it seems to make no sense, yet when it claims something is a prime it is indeed a prime. Unfortunately it misses pointing out that 3 is a prime, the final version that amatus came up with looks a bit like:
p is prime if sigma(2 .. p+1, (n ** (p+2)-1)/(n-1)) mod p) == p
While the second algorithm originally used Digital Roots to see if the sum of everything was equal to 1, amatus factored them out trying to create a slightly more general version so he could come up with a proof for it I suppose. You can still find the original DRb() version if you think of using b-1 in place of p. It also doesn't get trapped by Carmichael numbers, but it fails to find the prime number 3, so it might miss others. Finding a way to remove the sigma from this one is my current focus, that and finding a proof.
note: I might have typos in these as I am effectively plucking them from memory at the moment. I will look them over later when I find my notes and validate them.