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Math 2.0

Number theory Theorem

Zeckendorf's theorem

Builds on Fibonacci sequence

Statement

Let $F_k$ be the Fibonacci numbers, and let $N$ be a positive integer. Then there is a unique finite set $\{c_1, \dots, c_r\}$ of integers, with $r \ge 1$ , such that $N = F_{c_1} + F_{c_2} + \dots + F_{c_r},$ each $c_i \ge 2$ , and $c_{i+1} \ge c_i + 2$ for $1 \le i < r$ .

Remarks

The condition says that the sum uses distinct Fibonacci numbers and no two consecutive ones. The bound $c_i \ge 2$ excludes $F_0 = 0$ and the repeated value $F_1 = F_2 = 1$ . Example: $100 = 89 + 8 + 3 = F_{11} + F_6 + F_4$ . The greedy method finds the representation: subtract the largest Fibonacci number that is at most $N$ , and repeat.

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Fibonacci sequence example
Zeckendorf's theorem statement

Further reading

Web https://en.wikipedia.org/wiki/Zeckendorf's_theorem, lead section

About this page

Tag	`05DEJR6TGF`
URI	`m2://05DEJR6TGF`
Kind	statement, theorem
Writing	W1 stated
MSC	11B39
Status	This copy of the site has no build data. Run `m2 status 05DEJR6TGF`.
Uses	Fibonacci sequence
Same as	Wikidata Q1188392 Wikipedia Zeckendorf's theorem

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