Here are some known facts about the Fibonacci numbers and then some questions regarding them .
1.Carmichael's theorem : For every $n>12$ $F_n$ has a prime divisor which doesn't divide any of the numbers $F_m$ with $m<n$ .
(I'd like to see a proof of this but maybe it's too hard )
This has a striking similarity with Zsigmondy's theorem and this similarity makes much more sense when we think about the well-known formula for $F_n$: $$F_n=\frac{x^n-y^n}{\sqrt{5}}$$ (difference of powers like in Zsigmondy's but they're not integers )
2.The gcd property . It's well-known that : $$(F_n,F_m)=F_{(n,m)}$$
Looking again at the representation of $F_n$ we make the connection with the following :
$$(a^m-b^m,a^n-b^n)=a^{(n,m)}-b^{(n,m)}$$
As a corollary of this we can get : $$F_n \mid F_{kn}$$ which has the correspondent :
$$a^n-b^n \mid a^{kn}-b^{kn}$$
3.The fact that $$F_{p-\left( \frac{p}{5} \right )} \equiv 0 \pmod{p}$$ for $p$ a prime.
(I'd like to see a proof of this )
It seems like an equivalent to FLT in some way .Because of the difference of powers it seems connected with $a^{p-1}-b^{p-1} \equiv 0 \pmod{p}$
4.The Fibonacci binomial coefficients . The numbers : $$\binom{m+n}{n}_F=\frac{F_{m+n} \cdot \ldots \cdot F_{m+1}}{F_{n} \cdot \ldots \cdot F_1}$$ are always integers.
(I'd like to see a proof of this also ).
This seems to be a $q$-analog of the binomial coefficients .
Now I have three main questions :
What other such similarities exist between Fibonacci numbers and natural numbers?
Why these things really happen ?? I'd accept any proofs (for 1 , 3 or 4 ) but I am especially searching for some reasons of why things work exactly this way .Maybe someone can show us the big picture of why this stuff really works (hope you get what I mean ) .
What other sequences have such remarkable properties ? (ex what about Lucas's , Pell's or any other sequences ?)
I'd appreciate any answers to these questions .Thank you for all the help.
EDIT Another nice fact about Fibonacci numbers is this :
5.The lifting lemma . If $a \mid F_n$ then $a^{k+1} \mid F_{na^k}$ for every $k$ .
(found here An analogue of Hensel's lifting for Fibonacci numbers )
This is clearly related with the LTE lemma :
$$v_p(a^{nk}-b^{nk})=v_p(a^k-b^k) +v_p(n)$$
I have also a new question .
- Is it true that $v_p(F_{nk})=v_p(F_{k})+v_p(n)$ with $p$ prime which divides $F_k$ (the above lifting theorem for Fibonacci numbers would be a corollary of this )?
Is this true? (I have some doubts ) .What do you think ?
