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Zeckendorf ’s theorem∗
CompositeFan†
2013-03-21 20:50:00
Theorem. Every positive integer can be represented as a sum of distinct
non-consecutive Fibonacci numbers in a unique way.
This is Zeckendorf ’s theorem, first formulated by Edouard Zeckendorf.
For our purposes here, define the Fibonacci sequence thus: F0 = 1, F1 = 1
and Fm = Fm−2 + Fm−1 for all m > 0. 1 and 1 are not distinct even though the
first is F0 and the latter is F1 . We will consider two Fibonacci numbers Fi and
Fj consecutive if their indexes i and j are consecutive integers, e.g., j = i + 1.
A consequence of the theorem is that for every positive integer n there is a
unique ordered tuplet Z consisting of k elements, all 0s or 1s, such that
k
X
Zi Fi = n,
i=1
where Zi is the ith element in Z. This ordered tuplet Z is the Zeckendorf
representation of n, or we might even say the Fibonacci base representation of
n (or the Fibonacci coding of n).
So for example, 53 = 34 + 13 + 5 + 1, that is, F8 +F6 +F4 +F1 . Furthermore,
Z = (1, 0, 1, 0, 1, 0, 0, 1). We list the constituent elements in descending order
from Zk to Z1 to facilitate reinterpretation as a binary integer, 10101001 (or
169) in this example. Taking the Zeckendorf representations of integers in order
and reinterpreting in binary as
k
X
Zi 2i−1
i=1
gives the sequence 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, ... (A003714
in Sloane’s OEIS). It can be observed that these numbers have no consecutive
1s in their binary representations.
∗ hZeckendorfsTheoremi created: h2013-03-21i by: hCompositeFani version: h38120i
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References
[1] J. Tatersall, Elementary number theory in nine chapters Cambridge: Cambridge University Press (2005): 44
[2] J.-P. Allouche, J. Shallit and G. Skordev, “Self-generating sets, integers with
missing blocks and substitutions” Discrete Math., 292 (2005): 1 - 15
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