## Empty one year ago Prove that any number $$x \in \mathbb{N}$$ can be represented uniquely by the sum of unique Fibonacci numbers that are NOT consecutive.

1. ParthKohli

This is a thing?

2. Empty

I guess it gives it away to say this is called Zeckendorf's theorem. Rofl I just discovered this and thought it sounded cool.

3. ParthKohli

Things that are on my mind right now: - Induction - Creating various independent sequences of numbers that have no term in common and also that their union is $$\mathbb N$$

4. Empty

Let's just try to figure out this proof together here: https://en.wikipedia.org/wiki/Zeckendorf's_theorem#Proof

5. ParthKohli

Oh, great. I got the induction thing right.

6. Empty

I feel like understanding how someone could prove such a crazy statement as this would lead us to understanding some pretty interesting stuff haha.

7. ParthKohli

Strong-induction is really mind-boggling. It's amazing how well it works if you don't know how to prove things.

8. ParthKohli

How did they even come up with that proof? The uniqueness one is even worse.

9. ikram002p

-.-

10. Empty

Yeah this is pretty much too crazy for me, I wonder if numberphile did a video on this or if there's a youtube video of someone explaining it to me

11. ParthKohli

Math is cool.