# Fibonacci Mystery

A new friend was recently showing me an interesting mystery using the Fibonacci Numbers that I am at a loss to explain. I was hoping to post it here to see if someone else can help solve this mystery with a proof or explanation.

First, let's define n as an integer number greater than zero. And F(n) as the nth number of the Fibonacci series. In other words:

n | F(n) |
---|---|

**Table #1**

As you can see from above, the definition of a Fibonacci number is the sum of the two previous Fibonacci. This can be written as:

**F(n+2) = F(n+1) + F(n)**

This can be better shown as:

n | F(n) | F(n+1) | F(n+2) |
---|---|---|---|

**Table #2**

Now here is where an interesting pattern starts appearing that I hope someone can explain to me. Assume x is the multiplication of two Fibonacci numbers that only has a single Fibonacci number between them. This can be written as x = F(n) * F(n+2).

And assume y is the the middle Fibonacci number between the previous two Fibonacci numbers times itself. This can be written as y = F(n+1) ^ 2. Finally, z is the difference between x and y.

This can be better shown as:

n | F(n) | F(n+1) | F(n+2) | x = F(n) * F(n+2) | y = F(n+1) ^ 2 | z = x - y |
---|---|---|---|---|---|---|

**Table #3**

Notice that z alternates between -1 and 1.

Also, this is true:

n | z^2 = 1 | z = -1^(n+1) |
---|---|---|

**Table #4**

While this pattern is very interesting, it is not a proof. Can someone explain or prove to me why this is true?

Someone just emailed me this proof. Let me know if you find anything wrong with it, because I cannot.

Someone else, who also wishes to remain anonymous, sent in another proof using a totally different approach to proving the same thing. Again, if you find anything wrong with it, please let me know.

I already tried proving these equations; but so far, I've had no success.

or

I've even tried proving:

or

Thank you very much!

by Phil for Humanity

on 10/01/2011