- #1

Spider

- 1

- 0

Regards!

Conversely, any representation of the form (Zeckendorf's Theorem)

$$n=F_{k_{1}}+F_{k_{2}}+...+F_{k_{r}}, \quad k_{1}\gg k_{2}\gg ... \gg k_{r} \gg 0.$$

Implies that

$$ F_{k_{1}} \leq n < F_{k_{1}+1}, $$

because the largest possible value of $F_{k_{2}}+...+F_{k_{r}},$ when $k\gg k_{2}\gg...\gg k_{r}\gg0\:$is

$$F_{k-2}+F_{k-4}+...+F_{k\,mod\,2+2}=F_{k-1}-1, \quad \quad if\: k\geq2.$$

This last equation is to prove by induction on $k;$ the left-hand side is zero when $k$ is $2$ or $3$. Therefore $k_{1}$ is the greedily chosen value described earlier, and the representation must be unique.