Older blog entries for major (starting at number 1)

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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