pseudo-number theorist ramblings....

I just wanted to share and discuss some cool numeric patterns I've found playing around with the mod9 system. Pretty much all of them have already been found but its pretty cool trying to come up with applications utilising these patterns. I know alarm bells are already ringing with that thread name and my user name and the metatron's cube that makes up my avator so I just want to reasure you.
Relax, I'll try to keep the crazy shit to a minimum.

An operation that is used frequently in modular arithmetic is adding the constituent numbers of a double or greater digit number together until the sum is a single digit number. So 14 would be reduced to 5 and 64 would reduce to 1. This is what can be used to find patterns. Theres nothing hocus pocus about this despite the unfortunate superficial similarity to numerology. Numerologists apply digit reduction inconsistantly to confirm their beliefs and attach unwarrented significence to a digit value that reoccurs, two things that don't happen here.
Note: I'm dealing myself with only the base10 system but your welcome to find patterns on other
bases.

Ok from 1 start doubling, digitally reducing as you do so.
Doubling before digit reduction:
1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,1 6384,32768...
After reduction:
1,2,4,8,7(1+6),5(3+2),1(6+4=1+0),2(1+2+8=1+1),4(25 6=13)8(512),7(1024),5(2048=14),1(4096=19=10)....

So we find this repeating pattern of 1,2,4,8,7,5 that extends forward infinitly. The really amazing thing about this pattern is that it holds even in reverse with fractions! So by halfing from 1 we find:
1,0.5(5),0.25(7),0.125(8),0.0625(4),0.03125(2),0.0 15625(1)...

You might have noticed that 3 numbers are missing from this sequence: 3 (how apt), 6 and 9.
These numbers have quite interesting proporties as we'll see now:
Lets apply doubling to both 3 and 6 simultaneosly.

3,6,12(3),24(6),48(3),96(6),192(3)...
6,12(3),24(6),48(3),96(6),192(3),384(6)...

They seem to ossilate back and forth from each other.
Lets see if the pattern holds in reverse:

3-1.5(6),0.75(3),0.375(6),0.1875(3)...
It does!

Finally lets deal with 9:
9,18(9),36(9)72(9),144(9)..
9,4.5(9),2.25(9),1.125(9),0.5625(9)..
Basicly 9 never changes.

Now lets check out the fibonacci sequence:

Heres a list to the first 100 numbers.
Now watch what happens when we begin to reduce them.
1
1
2
3
5
8
4 (13)
3 (21)
7 (34)
1 (55)
8 (89)
9 (144)
8 (233)
8 (377)
7 (610)
6 (987)
4 (1597)
1 (2584)
5 (4181)
6 (6765)
2 (10946)
8 (17711)
1 (28657)
9 (46368)
1 (75025)
1 (121393)
2 (196418)
Etc etc etc......

The repeating pattern is this:
1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1,9
Now lets analyse some interesting aspects to this pattern.
First of all the two adjacent 1's are mirrored exactly half way into the pattern by adjacent 8's and that these doubles are preceeded by a 9. Lets put the pattern into two rows:
1,1,2,3,5,8,4,3,7,1,8,9
8,8,7,6,4,1,5,6,2,8,1,9

Add up the pairs of numbers in each of the twelve columns, what do you get?

1,1,2,3,5,8,4,3,7,1,8,9
8,8,7,6,4,1,5,6,2,8,1,9
-----------------------------
99999999999999999

I'm too sleepy to carry on but I might update this with more numeric crazies if there is actual interest. I'd like to add the disclaimer that I'm not a mathmatician and if I was, I'd be an incompetent one.

I can mail you a document about number theory if you'd like, which introduces modular arithmetic.

The key to modular arithmetic is that you consistently define a finite number of elements which together represent all integers. By which I mean, in mod 9, for instance, you only need to consider the symbols 0, 1, ... , 8. The rest are superfluous. What 8 here actually symbolises (hopefully UM won't enter this thread and discover mathematicians using the same symbol to mean different things, the concept is clearly too much for him) is [8] = { ... , 8 - 2*9, 8 - 9, 8, 8 + 9, 8 + 2*9, ... }, which is the set of all numbers with remainder 8 after division by 9. After a small amount of work, one can define addition and multiplication on such sets.

So, for doubling, we start with 1, then 2, then 4, then 8. But when we double 8 (once the small amount of work is done), it is completely legitimate to say the new element we get is 7 (inb4 modular arithmetic is a crock of shit). Then when we double 7 to get 5, and finally we double 5 to get 1.

In this interpretation, it becomes clear that all repeated multiplication sequences such as yours must eventually start repeating: there are a finite number of symbols, so eventually you must hit a previous symbol, and the process repeats.

The fact you can also do this backwards indefinitely is essentially because 2 is what we call a 'unit' in mod 9, which means there is a number u such that 2*u = 1. That number u turns out to be 5. This means to divide by 2, one may simply multiply by 5; say we want to divide x by 2. This means we want to find y where x = 2*y. Then 5*x = 5*2*y = 1*y = y.

Not all numbers are units. For example, 6 in mod 9. 0*6 = 0, 1*6 = 6, 2*6 = 3, 3*6 = 0, 4*6 = 6, 5*6 = 3. So we can't in general divide by 6.

That would be great Xei, thanks. I'll first try to grok the rest of your post.
Its interesting seeing how these simple rules create such interesting dynamics.
How does this explain the repeating pattern found in the fibonacci sequence though?
I would think that given its nature, a repeating pattern is suprising. I understand that since the fibonacci sequence is infinite, All numbers on it would share their digital roots (I think thats the right word) with an infinity of other numbers. I just don't understand why these repeating links are placed precisely every 24 places. I guess the interesting mirror symmetery of the number pattern itself can be explained away by the innate laws of modular arithmetic. I don't think I understand everything in your post so I'll probably practice creating my own examples other then the 8 to see if I can grok it.

I don't actually know that, although I imagine it's quite well studied and not so tricky to figure out after a bit of work. See if you can do it under the 'finite elements' interpretation, where you literally don't worry about what the Fibonacci number is in the integers, but just what it is in terms of the symbols 0, 1, ... , 8. In fact by writing that I just figured it out, it's quite easy given what I tried to explain above. See if you can get it. :P

PM me your email address and I'll send you the document. The relevant bit starts at section 4.1. But I really recommend reading it all the way through (except maybe section 3 first time), it's a perfect introduction to what it's like on a university maths course (it's from a first term lecture series), the pure half anyway; and you need literally zero prior knowledge, but once you're done you'll know more number theory than your average mathematics graduate. The stuff in section 6 is still probably the coolest thing I have seen in my degree, it's about the different sizes of infinity.

What y'all are discussing isn't real. :chuckle:

Sorry, I couldn't resist the opportunity. Carry on. Have a nice day.

I enjoy thinking about modulus as well, but not in the "clock arithmetic" sense.

To be honest 'clock arithmetic' is a pretty perfect analogy for modular arithmetic and a good thing to have in mind when you're doing it. What's your beef?

Quote:

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Originally Posted by Xei

What's your beef?

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Medium rare, with steak rub on a skillet.

Me too actually.

Xei I sent you my email but you still haven't sent me the papers.

Oh sorry I was expecting a PM. I'll check my mails now. :P