Every positive integer can be represented uniquely as a sum of one or more non-consecutive Fibonacci numbers. Proved by Edouard Zeckendorf, it gives a natural positional numeral system built from the Fibonacci sequence.
Facts
StatementEvery positive integer can be represented uniquely as a sum of one or more Fibonacci numbers such that the sum includes no two consecutive Fibonacci numbers; this sum is the number's Zeckendorf representation. 2 Proof Method Proof Method Classification
Statement Form Connections
Has Statement Form
In Branch
Principal Result Of
Source Edouard Zeckendorf (Wikipedia)
Proved By
Source Zeckendorf's Theorem (Wikipedia)
Sources
1. Wikipedia: Zeckendorf's theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
It 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.
2. Zeckendorf's Theorem (Wikipedia)
Wikimedia FoundationReferences section
Zeckendorf, E. (1972). "Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas". Bull. Soc. R. Sci. Liège (in French). 41: 179-182.
Proved By: Edouard Zeckendorf, Lead sentence
Zeckendorf's theorem, named after amateur mathematician Edouard Zeckendorf, is a result about the representation of integers as sums of Fibonacci numbers.
View the Source 3. ProofWiki: Zeckendorf's Theorem
ProofWikiProof section
Aiming for a contradiction, suppose $S$ and $T$ are distinct Zeckendorf representations for $n$.
Proof section, proof method
We will use strong induction on $n$.
View the Source Edouard Zeckendorf (Wikipedia)
Principal Result Of: Edouard Zeckendorf, LeadQuote, Principal Result Of: Edouard Zeckendorf, Lead
now known as Zeckendorf's theorem: every positive whole number is either a Fibonacci number or can be written as a sum of distinct non-consecutive Fibonacci numbers
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