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


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 different from 3, then
is prime. (The converse is false.)
Algebraic identities
for n=1,2,...

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:

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

The series is convergent for
Obviously:

hence:

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