[Video] Deriving Binet's formula for the Fibonacci numbers

One of those formulas every mathematician loves ( I think ):

$$F_n = \frac{1}{\sqrt{5}}( \phi^n - (1-\phi)^n )$$

Here's how MathDoctorBob explains it.



Although I like The Doctor's videos ( I wished the real doctor would show up accusing me for abusing his name but taking me for a ride in his phone box anyway ), I wouldn't like to have Doctor Bob as a tutor in class, I simply wouldn't be able to catch up and I am not the audible type anyway. I like to read a bit, play and think a bit, read a bit, and so on. Every person has its own unique style of learning that works for him. Part of studying is discovering your own learning style.

Oh, and I think this formula beats the one of Binet ( although strictly speaking not in closed form ), because it fascinates me that the Fibonacci numbers are actually -in- the triangle of Pascal.

$$F_{n+1} = \sum_{k=0}^{n} {n-k \choose k}$$

Fibonacci series modulo m

For a while I thought I had discovered some new mathematics. Alas no. Every Coin Has Two Sides. This paper from 1960 by D.D. Wall has no secrets for me. I was well on my way re-discovering and proving until I searched for similar results. I used the Open University library services of course and within a few minutes I found a relevant paper. All this activity is due to NT book 2 from M381.

A Fibonacci series modulo m is cyclic. For example the series mod 3, starting with 0,1 is :
0 1 1 2 0 2 2 1 - 0 1 etc.
and has length 8.

More in the paper.

P.S.
Dream on. I wished that I could travel back in time. To the year 1914 for example. I would go to Cambridge, England. I would take Number Theory lectures from G.H. Hardy and would make friends with Srinivasa Ramanujan. On the way back to 2011 I would stop the clock somewhere in the midst of World War II, to meet Alan Turing to watch him cracking the Enigma code.