Fibonacci number

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In mathematics, the Fibonacci numbers form a sequence in which the first number is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers in the series. In mathematical terms, it is defined by the following recurrence relation:

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

It was first used to represent the growth of a colony of rabbits, starting with a single pair of rabbits. It has applications in mathematics as well as other sciences, and is a popular illustration of recursive programming in computer science.

Divisibility properties

We will apply the following simple observation to Fibonacci numbers:

if three integers   satisfy equality   then


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 different from 3, then   is prime. (The converse is false.)

Algebraic identities

  •     for n=1,2,...

Direct formula and the golden ratio

We have

for every .

Indeed, let    and   .  Let


  •     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:

Fibonacci generating function

The Fibonacci generating function is defined as the sum of the following power series:

The series is convergent for    Obviously:


Value   is a rational number whenever x is rational. For instance, for x = ½:

and for x = −½ (after multiplying the equality by −1):

Further reading