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

The sequence of Fibonacci numbers starts: 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 different from 3, 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