FIBONACCI NUMBERSAND

LUCAS NUMBERS

Fibonacci

Numbers

INTRODUCTION

In mathematics, the Fibonacci numbers or Fibonacci series are the numbers in the following integer sequence:

1, 1, 2, 3, 5, 8, 13, 21 ,34, 55…

or (in modern usage):

0 ,1, 1, 2, 3, 5, 8, 13, 21 ,34, 55…

In mathematical terms,

Fn=Fn-1 + Fn-2 ,

with seed values :

F0=1 and F1=1

or

F0=0 and F1=1

…

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610

The first 15 Fibonacci numbers Fn for n = 0, 1, 2, ..., 14 are:

The sequence can also be extended to negative index n :

Fn-2 = Fn – Fn-1

which yields the sequence of "negafibonacci“ numbers satisfying:

F-n = (-1)n+1 Fn

Thus the bidirectional sequence is:

F−7 F−6 F−5 F−4 F−3 F−2 F−1 F0 F1 F2 F3 F4 F5 F6 F7

13 −8 5 −3 2 −1 1 0 1 1 2 3 5 8 13

HISTORY

The Fibonacci sequence is named after Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics.

Matrix

Form

Let L be the linear operator on R represented by the matrix

For any vector v = (x, y) , we have that

In particular, for the vector uk whose coordinates are two consecutive Fibonacci numbers (Fk, Fk−1) , we have that

=

Thus we can produce a vector whose coordinates are two consecutive Fibonacci numbers, by applying L many times to the vector u1 with coordinates (F1, F0) =(1,0)

Equation above is nothing but a reformulation of the definition of Fibonacci numbers. This equation, however, allows us to find an explicit formula for Fibonacci numbers as soon as we know how to calculate the powers An of the matrix A with the help of the diagonalization.

DIAGONALIZATIONAny square matrix A is dioganalizable if there is a non-singular matrix S of the same size such that the matrix S−1AS is diagonal. That means all entries of S−1AS except possibly diagonal entries are zeros. The numbers which show up on the diagonal of S−1AS are the eigenvalues of A. For a diagonal matrix, it is very easy to calculate its powers.

Let A be a dioganalizable matrix of size m × m, and assume that

for a non-singular matrix S. Then for an integer n ≥ 0

Hence, An = SΛnS−1 where Λ is denote diagonal matrix with Eigen values.

Back to our problem…we begin with finding the eigenvalues of A as the roots of its characteristic polynomial

Solving the above quadratic equation we get,

since the two eigenvalues are real and distinct, that the matrix A dioganalizable. Now let us find the eigenvectors from the equations

We solve these equations and find eigenvectors:

Now the non singular matrix S

And

We know that An = SΛnS−1

Using the reformulated equation

Equating the entries of the vectors in the last formula we obtain

KEPLER THEOREM

Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that the limit approaches the golden ratio .

Now lets prove kepler’s theorem…

RECOGNIZING FIBONACCI NUMBERS

The question may arise whether a positive integer x is a Fibonacci number. This is true if and only if one or both of 5x2 + 4 or 5x2 - 4 is a perfect square.

Example-

Q-Whether 5 is a Fibonacci number or not?

A- Yes it is, because

5(5*5) - 4= 121

And √121 = 11

DIVISIBILITY PROPERTIESEvery 3rd number of the sequence is even and more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property :

A Fibonacci prime is a Fibonacci number that is prime. The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...

FIBONACCI PRIMES

APPLICATIONS

• Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems.

• They also appear in biological settings, such as branching in trees, the fruit sprouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.

OBSERVATION

The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle:

Lucas Numbers

INTRODUCTION

Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms. The first two Lucas numbers are L0 = 2 and L1 = 1 as opposed to the first two Fibonacci numbers F0 = 0 and F1 = 1. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.

2, 1, 3, 4 , 7, 11, 18, 29, 47, 76, 123, …

…

Using Ln−2 = Ln − Ln−1, one can extend the Lucas numbers to negative integers to obtain a doubly infinite sequence:

..., −11, 7, −4, 3, −1, 2, 1, 3, 4, 7, 11, ... (terms Ln   for -6<n<6 are shown).

The formula for terms with negative indices in this sequence is :

L-n = (-1)n+1 Ln

RELATIONSHIP TO FIBONACCI NUMBERSThe Lucas numbers are related to the Fibonacci numbers by the identities :

LUCAS PRIMES

A Lucas prime is a Lucas number that is prime. The first few Lucas primes are

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, ...