> 
> I am looking for a proof on why 
>  $$\frac{1}{\sqrt 5}\left[\left(\frac{1+\sqrt 5}{2}\right)^n-\left(\frac{1-\sqrt 5}{2}\right)^n\right]$$
>  an integer.
> 
> 
> 

I have seen many proofs on this, but they all refer to a properties of Fibonacci numbers, which shouldn't be necessary.

I am trying to see why it is true using purely elementary results such as the binomial formula. Clearly this reduces to 
$$\frac{1}{2^n\sqrt 5}\sum\_{k=0}^n {n\choose k }\left(1-(-1)^k\right) 5^{k/2}$$ 
I am looking for a "divisibility" argument to see why this is an integer.