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Name: Mark Ferrell
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30 Sep 2005 (updated 30 Sep 2005 at 19:16 UTC) »

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.

24 Sep 2003 (updated 24 Sep 2003 at 04:54 UTC) »

I have found numerous references to some number theory that seems to draw a parallel to my whole computing sum's scenerio that has been driving me crazy as of recent.

The key term is digital root. It seems that my reducing the values of a number to a single digit by adding together each of the digits in the number is computing the digital root. Okay, no biggi there. There also seems to be a semi-weak relation to sigma codes, though I havn't quite decided yet if what I have run across is infact some property of sigma codes or not.

Originally I had used a double itterative loop to compute the digital root, but I have run across a much faster way of computing it for base10 (the 9 can be replaced with baseN - 1):

dr = 1 + (((n ** x) - 1) % 9)

And I have run across a curious conjecture in my recent insanity into this whole thing:

dr(n ** x) = dr(n ** dr(x))

It also seems that: if dr(n) == dr(i) then dr(n ** x) = dr(i ** x)

I am still trying to piece together a proof for this with a friend at work. As soon as we come up with one (more likely he will do all the work here, I am horrible at proofs), we have decided to see about finding a proof that describes the reproducable patterns that occure. And then it is off to explain why it is that predictable powers in the pattern intersect at a digital root of 1 at the same time, and a seperate predictable set of powers never reduce 1, and why it is that these powers are always a fixed distance apart.

19 Sep 2003 (updated 19 Sep 2003 at 02:27 UTC) »

Okay, so I managed to get dragged out of my cave I dwell in and into the world again, tossed at the doorsteps to this site.

The trust metric is a curious concept. I recognize the reasons for using it, and at the same time I worry. In the end I guess it all comes down to how open the users are to certifying people who have different points of view, potentially conflicting ones. Only time will tell I guess, though it might be interesting to graph out.

On that note, I ran across a curious pattern the other night, summing up the values of each of the values of base2 powers. i.e.

1 = 1, 2 = 2, 4 = 4, 8 = 8, 16 = 1 + 6 = 7, 32 = 3 + 2 = 5, 64 = 6 + 4 = 10 = 1 + 0 = 1

So on and so forth. There is a pattern that shows up if you carry it out far enough.

1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5 ...

The pattern continues on indeffinately it seems, or at least it did when computing to 2^1000. A similar pattern arives for base4 when representing the sums in base10, 1 4 7 1 4 7, and base5 is the inverse of base2, 5 7 8 4 2 1. Base7 seems to be the inverse of base 4, 7 4 1 7 4 1. Base3, 6, and 9 all end up sum'ing to 9 over and over again (with the exception of the first couple of powers), base 8 to 1 8 1 8 1 8.

If you go higher up the bases, like base31, you find the same pattern as base4 (3 + 1 = 4, odd coincidence), and base23 has the same pattern as base5 (2+3?). In the end, everything above base9 generated a pattern that matched one of the base1-9 patterns.

The only bases to not generate an infinite list of 9's where base 1, 2, 4, 8, 7 and 5 patterns, which are the same numbers for base2 and base5. I have been meaning to go see if such a mundane thing as reducing the sums in such a way was ever noticed before and documented, but I keep forgetting to. Maybe this entry will serve as a reminder. It all graphs out very curiously, though for the most part it seems to be no more usefull then helping one go to sleep at night.


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