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

Question

parthkohli · Answer

This is a thing?

empty · Answer

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

parthkohli · Answer

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 $\mathbb N$

empty · Answer

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

parthkohli · Answer

Oh, great. I got the induction thing right.

empty · Answer

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

parthkohli · Answer

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

parthkohli · Answer

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

ikram002p · Answer

-.-

empty · Answer

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

parthkohli · Answer

Math is cool.