Fibonacci number

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In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation:

The sequence of fibonacci numbers start: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

The sequence of Fibonacci numbers was first used to represent the growth of a colony of rabbits, starting with a single pair of rabbits.


Properties

We will apply the following simple observation to Fibonacci numbers:

if three integers   satisfy equality   then


Indeed,

and the rest is an easy induction.


for all integers   such that


Indeed, the equality holds for   and the rest is a routine induction on

Next, since ,  the above equality implies:

which, via Euclid algorithm, leads to:


Let's note the two instant corollaries of the above statement:


  • If   divides then divides
  • If   is a prime number then   is prime. (The converse is false.)


Direct formula and the golden ratio

We have

for every .

Indeed, let    and   .  Let

Then:

  •     and    
  •     hence    
  •     hence    

for every . Thus   for every and the formula is proved.

Furthermore, we have:

It follows that

  is the nearest integer to 

for every . The above constant   is known as the famous golden ratio   Thus:

Further reading