Count up with Zeckendorf's theorem!

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Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 3:31 am UTC

Wikipedia wrote:Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.


1 = 1
2 = 10
3 = 100
4 = 101
5 = 1000
6 = ...?



Just for reference, and those unfamiliar with the Fibonacci sequence who want to play, the first few Fibonacci numbers are:

1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711

That should do us just fine, for now... :twisted:
Last edited by Sean Quixote on Wed Sep 28, 2011 8:29 pm UTC, edited 1 time in total.
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Re: Count up with Zeckendorf's theorem!

Postby curtis95112 » Wed Sep 14, 2011 2:25 pm UTC

Since you posted up to 5, I'll start with 6.

6 = 111
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 3:00 pm UTC

7 = 1010, if I'm understanding this correctly.
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Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 3:27 pm UTC

Looks like 6 should have been 1001 since the Fibonacci numbers cannot be consecutive (i.e. your number should never have 2 ones next to each other).

8 = 10000
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 5:48 pm UTC

9 = 10001

That's right, a-wan. Basically the way I see it, what's going to go on here is we're gonna pretend like we're translating Zeckendorf's representation into a base system that resembles binary, in that it will contain only ones and zeros. The Fibonacci numbers will be our "orders of magnitude" or place values: from the rightmost digit, just go in your head, "1, 2, 3, 5, 8, 13, 21, 34, 55, etc..."
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 6:54 pm UTC

10 = 10010

This doesn't look like a very efficient number system for small numbers. I'm sure it gets better for larger ones.
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 7:08 pm UTC

11 = 10100

Yeah, not really. Never even though of that before, but yeah, not really. :P I guess, every time we reach another Fibonacci number, the efficiency ratio will improve by approximately a factor of phi?
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Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 7:25 pm UTC

12 = 10101
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 7:28 pm UTC

13 = 100000
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Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 7:31 pm UTC

14 = 100001
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 7:32 pm UTC

15 = 100010
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Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 7:52 pm UTC

16 = 100100
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 8:04 pm UTC

17 = 100101

Ah, 17... Anyone wanna take a gander as to what it has in common with 72, 305, 1292, 5473, et cetera?
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 8:15 pm UTC

18 = 101000

At a guess and a look at a couple of them, it looks like they're all 100...101
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 8:23 pm UTC

19 = 101001

Eh.. maybe. I dunno actually, because the answer I was looking for technically has little if anything to do with the Zeckendorf's representation. It has more to do with another thing that I came up with, but I never was sure what I should call it...
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Re: Count up with Zeckendorf's theorem!

Postby a-wan » Wed Sep 14, 2011 9:00 pm UTC

20 = 101010
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 9:32 pm UTC

21 = 1000000

Sean Quixote wrote:Ah, 17... Anyone wanna take a gander as to what it has in common with 72, 305, 1292, 5473, et cetera?

If you don't want a spoiler alert: You might not want to read my thread over in the math forum. ;)
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 9:57 pm UTC

22 = 1000001
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 10:00 pm UTC

23 = 1000010
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Wed Sep 14, 2011 10:07 pm UTC

24 = 1000100
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Wed Sep 14, 2011 10:12 pm UTC

25 = 1000101
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Thu Sep 15, 2011 3:53 am UTC

26 = 1001000
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Thu Sep 15, 2011 1:15 pm UTC

27 = 1001001
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Thu Sep 15, 2011 6:16 pm UTC

28 = 1001010
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Thu Sep 15, 2011 6:21 pm UTC

29 = 1010000

I've never actually written this stuff out before (and now I guess I shouldn't have to ;)) so I just realized another thing that's going on here: if someone came along one day and said, "I want to create a base system that only has two symbols (1 and 0), but let's say that numbers can only be written in such a way that the 1s never touch eachother..." This is also what you would come up with. :roll:
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Fri Sep 16, 2011 3:36 am UTC

30 = 1010001
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Fri Sep 16, 2011 3:55 am UTC

31 = 1010010
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Fri Sep 16, 2011 5:06 am UTC

32 = 1010100
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Fri Sep 16, 2011 2:05 pm UTC

33 = 1010101
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Fri Sep 16, 2011 3:09 pm UTC

34 = 10000000
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Re: Count up with Zeckendorf's theorem!

Postby a-wan » Fri Sep 16, 2011 5:05 pm UTC

35 = 10000001
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Fri Sep 16, 2011 5:59 pm UTC

36 = 10000010
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Fri Sep 16, 2011 6:43 pm UTC

37 = 10000100
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Fri Sep 16, 2011 11:04 pm UTC

38 = 10000101
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Sat Sep 17, 2011 5:32 pm UTC

39 = 10001000
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Sat Sep 17, 2011 8:50 pm UTC

40 = 10001001
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Re: Count up with Zeckendorf's theorem!

Postby Anonymously Famous » Sat Sep 17, 2011 10:44 pm UTC

41 = 10001010
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Sun Sep 18, 2011 1:19 am UTC

42 = 10010000
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Re: Count up with Zeckendorf's theorem!

Postby gaga654 » Sun Sep 18, 2011 3:02 am UTC

43 = 10010001
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Re: Count up with Zeckendorf's theorem!

Postby Sean Quixote » Sun Sep 18, 2011 3:23 am UTC

44 = 10010010
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