# Zeckendorf’s theorem

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: $F_{0}=1$, $F_{1}=1$ and $F_{m}=F_{m-2}+F_{m-1}$ for all $m>0$. 1 and 1 are not distinct even though the first is $F_{0}$ and the latter is $F_{1}$. We will consider two Fibonacci numbers $F_{i}$ and $F_{j}$ 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

 $\sum_{i=1}^{k}Z_{i}F_{i}=n,$

where $Z_{i}$ is the $i$th element in $Z$. This ordered tuplet $Z$ is the 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, $F_{8}+F_{6}+F_{4}+F_{1}$. Furthermore, $Z=(1,0,1,0,1,0,0,1)$. We list the constituent elements in descending order from $Z_{k}$ to $Z_{1}$ 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

 $\sum_{i=1}^{k}Z_{i}2^{i-1}$

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.

## 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” , 292 (2005): 1 - 15
 Title Zeckendorf’s theorem Canonical name ZeckendorfsTheorem Date of creation 2013-03-22 16:03:57 Last modified on 2013-03-22 16:03:57 Owner CompositeFan (12809) Last modified by CompositeFan (12809) Numerical id 11 Author CompositeFan (12809) Entry type Theorem Classification msc 11A63 Classification msc 11B39 Synonym Zeckendorff’s theorem Related topic FibonacciSequence Related topic UniquenessOfDigitalRepresentation Defines Zeckendorf representation Defines Fibonacci base Defines Fibonacci coding