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:
31.5(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.


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