Zeckendorf's theorem

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.

Explanations

L1 human

Each positive integer is a sum of different Fibonacci numbers, no two of them neighbours in the sequence, in exactly one way. For example, $20 = 13 + 5 + 2$.