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[[Image:Zeckendorf representations.png|thumb|right|320px|The first 160 integers (on the X-axis) broken down into Zeckendorf representation. Each rectangle's color corresponds with a Fibonacci number and its height corresponds with the number's value.]]
'''Zeckendorf's theorem''', named after [[Belgium|Belgian]] [[mathematician]] [[Edouard Zeckendorf]], is a [[theorem]] about the representation of [[integer]]s as sums of [[Fibonacci number]]s.


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.  More precisely, if {{mvar|N}} is any positive integer, there exist positive integers {{math|''c''<sub>''i''</sub> ≥ 2}}, with {{math|''c''<sub>''i'' + 1</sub> > ''c''<sub>''i''</sub> + 1}}, such that


:<math>N = \sum_{i = 0}^k F_{c_i},</math>
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where {{mvar|F<sub>n</sub>}} is the {{mvar|n}}<sup>th</sup> Fibonacci number. Such a sum is called the '''Zeckendorf representation''' of {{mvar|N}}. The [[Fibonacci coding]] of {{mvar|N}} can be derived from its Zeckendorf representation.
 
For example, the Zeckendorf representation of 100 is
 
:{{math|1=100 = 89 + 8 + 3}}.
 
There are other ways of representing 100 as the sum of Fibonacci numbers&nbsp;– for example
 
:{{math|1=100 = 89 + 8 + 2 + 1}}
:{{math|1=100 = 55 + 34 + 8 + 3}}
 
but these are not Zeckendorf representations because 1 and 2 are consecutive Fibonacci numbers, as are 34 and 55.
 
For any given positive integer, a representation that satisfies the conditions of Zeckendorf's theorem can be found by using a [[greedy algorithm]], choosing the largest possible Fibonacci number at each stage.
 
==Proof==
{{unsourced section|date=September 2012}}
Zeckendorf's theorem has two parts:
 
#'''Existence''': every positive integer&nbsp;{{mvar|n}} has a Zeckendorf representation.
#'''Uniqueness''': no positive integer&nbsp;{{mvar|n}} has two different Zeckendorf representations.
 
The first part of Zeckendorf's theorem (existence) can be proven by [[Mathematical induction|induction]]. For {{math|1=''n'' = 1, 2, 3}} it is clearly true (as these are Fibonacci numbers), for {{math|1=''n'' = 4}} we have {{math|1=4 = 3 + 1}}. Now suppose each {{math|''n'' ≤ ''k''}} has a Zeckendorf representation. If {{math|''k'' + 1}} is a Fibonacci number then we're done, else there exists {{mvar|j}} such that {{math|''F''<sub>''j''</sub> < ''k'' + 1 < ''F''<sub>''j'' + 1</sub> }}. Now consider {{math|1=''a'' = ''k'' + 1 − ''F''<sub>''j''</sub> }}. Since {{math|''a'' ≤ ''k''}}, {{mvar|a}} has a Zeckendorf representation (by the induction hypothesis). At the same time, {{math|''F''<sub>''j''</sub> + ''a'' < ''F''<sub>''j'' + 1</sub>}} and since {{math|''F''<sub>''j'' + 1</sub> {{=}} ''F''<sub>''j''</sub> + ''F''<sub>''j'' − 1</sub>}} (by definition of Fibonacci numbers), {{math|''a'' < ''F''<sub>''j'' − 1</sub>}}, so the Zeckendorf representation of {{mvar|a}} does not contain {{math|''F''<sub>''j'' − 1</sub> }}. As a result, {{math|''k'' + 1}} can be represented as the sum of {{mvar|F<sub>j</sub>}} and the Zeckendorf representation of {{mvar|a}}.
 
The second part of Zeckendorf's theorem (uniqueness) requires the following lemma:
 
:''Lemma'': The sum of any non-empty set of distinct, non-consecutive Fibonacci numbers whose largest member is {{mvar|F<sub>j</sub>}} is strictly less than the next largest Fibonacci number {{math|''F''<sub>''j'' + 1</sub> }}.
 
The lemma can be proven by induction on {{mvar|j}}.
 
Now take two non-empty sets of distinct non-consecutive Fibonacci numbers {{math|'''S'''}} and {{math|'''T'''}} which have the same sum. Consider sets {{math|'''S<var>′</var>'''}} and {{math|'''T<var>′</var>'''}}<!-- the use of <var>′</var> instead of ''′'' is not a necessity if the CSS defines "span.texhtml i" properly, along with "span.texhtml var". see [[template:math]] --> which are equal to {{math|'''S'''}} and {{math|'''T'''}} from which the common elements have been removed (i.e. {{math|1='''S<var>′</var>''' = '''S'''\'''T'''}} and {{math|1='''T<var>′</var>''' = '''T'''\'''S'''}}).  Since {{math|'''S'''}} and {{math|'''T'''}} had equal sum, and we have removed exactly the elements from {{math|'''S''' <!---->}}<math>\cap</math>{{math| '''T'''}} from both sets, {{math|'''S<var>′</var>'''}} and {{math|'''T<var>′</var>'''}} must have the same sum as well.
 
Now we will show [[Proof by contradiction|by contradiction]] that at least one of {{math|'''S<var>′</var>'''}} and {{math|'''T<var>′</var>'''}} is empty. Assume the contrary, i.e. that {{math|'''S<var>′</var>'''}} and {{math|'''T<var>′</var>'''}} are both non-empty and let the largest member of {{math|'''S<var>′</var>'''}} be {{mvar|F<sub>s</sub>}} and the largest member of {{math|'''T<var>′</var>'''}} be {{mvar|F<sub>t</sub>}}. Because {{math|'''S<var>′</var>'''}} and {{math|'''T<var>′</var>'''}} contain no common elements, {{math|''F''<sub>''s''</sub> ≠ ''F''<sub>''t''</sub>}}. [[Without loss of generality]], suppose {{math|''F''<sub>''s''</sub> < ''F''<sub>''t''</sub>}}. Then by the lemma, the sum of {{math|'''S<var>′</var>'''}} is strictly less than {{math|''F''<sub>''s'' + 1</sub>}} and so is strictly less than {{mvar|F<sub>t</sub>}}, whereas the sum of {{math|'''T<var>′</var>'''}} is clearly at least {{mvar|F<sub>t</sub>}}. This contradicts the fact that {{math|'''S<var>′</var>'''}} and {{math|'''T<var>′</var>'''}} have the same sum, and we can conclude that either {{math|'''S<var>′</var>'''}} or {{math|'''T<var>′</var>'''}} must be empty.
 
Now assume (again without loss of generality) that {{math|'''S<var>′</var>'''}} is empty. Then {{math|'''S<var>′</var>'''}} has sum 0, and so must {{math|'''T<var>′</var>'''}}. But since {{math|'''T<var>′</var>'''}} can only contain positive integers, it must be empty too. To conclude: {{math|1='''S<var>′</var>''' = '''T<var>′</var>''' = <!---->}}<math>\emptyset</math> which implies {{math|1='''S''' = '''T'''}}, proving that each Zeckendorf representation is unique.
 
==Fibonacci multiplication==
One can define the following operation <math>a\circ b</math> on natural numbers {{mvar|a}}, {{mvar|b}}:  given the Zeckendorf representations
<math>a=\sum_{i=0}^kF_{c_i}\;(c_i\ge2)</math> and <math>b=\sum_{j=0}^lF_{d_j}\;(d_j\ge2)</math> we define the '''Fibonacci product''' <math>a\circ b=\sum_{i=0}^k\sum_{j=0}^lF_{c_i+d_j}.</math>
 
For example, the Zeckendorf representation of 2 is <math>F_3</math>, and the Zeckendorf representation of 4 is <math>F_4 + F_2</math> (<math>F_1</math> is disallowed from representations), so <math>2 \circ 4 = F_{3+4} + F_{3+2} = 13 + 5 = 18.</math>
 
A simple rearrangement of sums shows that this is a [[commutative]] operation; however, [[Donald Knuth]] proved the surprising fact that this operation is also [[associative]].
 
==Representation with negafibonacci numbers==
The Fibonacci sequence can be extended to negative index&nbsp;{{mvar|n}} using the re-arranged recurrence relation
:<math>F_{n-2} = F_n - F_{n-1}, \, </math>
which yields the sequence of "[[negafibonacci]]" numbers satisfying
:<math>F_{-n} = (-1)^{n+1} F_n. \, </math>
 
Any integer can be uniquely represented<ref>Knuth, Donald. "Negafibonacci Numbers and the Hyperbolic Plane" Paper presented at the annual meeting of the Mathematical Association of America, The Fairmont Hotel, San Jose, CA. 2008-12-11 <http://www.allacademic.com/meta/p206842_index.html></ref> as a sum of negafibonacci numbers in which no two consecutive negafibonacci numbers are used.  For example:
 
* {{math|1=−11 = ''F''<sub>−4</sub> + ''F''<sub>−6</sub> = (−3) + (−8)}}
* {{math|1=12 = ''F''<sub>−2</sub> + ''F''<sub>−7</sub> = (−1) + 13}}
* {{math|1=24 =  ''F''<sub>−1</sub> + ''F''<sub>−4</sub> + ''F''<sub>−6</sub> + ''F''<sub>−9</sub> = 1 + (−3) + (−8) + 34}}
* {{math|1=−43 = ''F''<sub>−2</sub> + ''F''<sub>−7</sub> + ''F''<sub>−10</sub> = (−1) + 13 + (−55)}}
* 0 is represented by the [[empty sum]].
 
Note that {{math|1=0 = ''F''<sub>−1</sub> + ''F''<sub>−2</sub> }}, for example, so the uniqueness of the representation does depend on the condition that no two consecutive negafibonacci numbers are used.
 
This gives a [[NegaFibonacci_coding|system]] of coding [[integers]], similar to the representation of Zeckendorf's theorem. In the string representing the integer&nbsp;{{mvar|x}}, the {{mvar|n}}<sup>th</sup> digit is 1 if {{mvar|F<sub>n</sub>}} appears in the sum that represents {{mvar|x}}; that digit is 0 otherwise.   For example, 24 may be represented by the string 100101001, which has the digit 1 in places 9, 6, 4, and 1, because {{math|1=24 = ''F''<sub>−1</sub> + ''F''<sub>−4</sub> + ''F''<sub>−6</sub> + ''F''<sub>−9</sub> }}.  The integer&nbsp;{{mvar|x}} is represented by a string of odd length [[if and only if]] {{math|''x'' > 0}}.
 
==See also==
* [[Fibonacci coding]]
* [[Ostrowski numeration]]
 
==References==
{{reflist}}
* {{cite journal | first=Donald E. | last=Knuth | authorlink=Donald Knuth | title=Fibonacci multiplication |journal=Applied Mathematics Letters |volume=1 |issue=1 |year=1988 |pages=57–60 | doi=10.1016/0893-9659(88)90176-0 | zbl=0633.10011 | issn=0893-9659 }}
* {{cite journal | zbl=0252.10011 | last=Zeckendorf | first=E. | title=Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas | language=French | journal=Bull. Soc. R. Sci. Liège | volume=41 | pages=179-182 | year=1972 | issn=0037-9565 }}
 
==External links==
*{{MathWorld |title=Zeckendorf's Theorem |urlname=ZeckendorfsTheorem}}
*{{MathWorld |title=Zeckendorf Representation |urlname=ZeckendorfRepresentation}}
*[http://www.cut-the-knot.org/ctk/OnePile.shtml#fibnim Zeckendorf's theorem] at [[cut-the-knot]]
*{{SpringerEOM|urlname=Z/z120020|title=Zeckendorf representation|author=G.M. Phillips}}
*{{SloanesRef |sequencenumber=A101330|name=Knuth's Fibonacci (or circle) product}}
 
{{PlanetMath attribution|id=8810|title=proof that the Zeckendorf representation of a positive integer is unique}}
 
[[Category:Fibonacci numbers]]
[[Category:Theorems in number theory]]
[[Category:Articles containing proofs]]
 
[[de:Fibonacci-Folge#Zeckendorf-Theorem]]

Latest revision as of 08:51, 23 December 2014


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