FIBONACCI SERIES

 

The pattern of the "numbers of life" is elegantly simple. In the Fibonacci sequence, every number (after the first two) is the sum of the two preceding numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, and so on.

This looks like a simple pattern, yet it determines the shape of a mollusk's shell and a parrot's beak, or the sprouting of leaves from the stem of any plant--a revelation as surprising to us, as it probably was to Leonardo Pisano--later known as Fibonacci--almost 800 years ago. Pisano, the first great mathematician of medieval Europe, discovered these magical numbers by analyzing the birth rate of rabbits (More Details in our Bio section).

In addition to its connections with the natural world, the Fibonacci sequence has a number of curious mathematical properties. Perhaps the most amazing is that, as you proceed along the sequence, the ratios of the successive terms get closer and closer to the famous "golden ratio" number 1.61803 . . . , the "perfect proportion" ratio much beloved by the ancient Greeks. (Scroll down for further details).

Therefore, if Fn is the nth term in the Fibonacci Sequence, then Fn+2=Fn+1+Fn or Fn=Fn-1+Fn-2. Following this formula will result in the Fibonacci sequence.

If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ...) and we divide each by the number before it, we will find the following series of numbers:

1/1 = 1,   2/1 = 2,   3/2 = 1·5,   5/3 = 1·666...,   8/5 =

1·6,   13/8 = 1·625,   21/13 = 1·61538...

It is easier to see what is happening if we plot the ratios on a graph:

 

 

The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1·618034.

As we may observe from the Figure 2, the ratio f(n-1)/f(n) is 0,618033989... which is the reciprocal of the golden ratio.

  

Compute any number in the Fibonacci Series easily!

You can use phi to compute the nth number in the Fibonacci series (fn):

fn =  Phi n / 5½

(This provides an estimate which always rounds to the correct Fibonacci number.)

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Squares in the Fibonacci Series

F(1) = F(2) = 1 and F(12) = 144.

A positive integer n is always either odd or even, so n = 2k or n = 2k-1 for certain integer k. Then n2 = 4k2 or n2 = 4k2-4k+1. Thus, if we divide n2 by 4, the remainder will be either 0 or 1. Apparently a square can never have a remainder 2 or 3 when it is divided by 4.  Dividing the Fibonacci numbers by 4, the remainders are: 
1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, ...

The numbers 1, 1, 2, 3, 1, 0 are repeated again and again.

Thus, it is concluded from  this argument that no Fibonacci number of the form F(6k-3) or of the form F(6k-2) is a square because when a square is divided by 4, the remainder is 2 or 3.
 

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Fibonacci Primes

 

The first few Fibonacci numbers that are also prime numbers  are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …. It seems likely that there are infinitely many Fibonacci Primes, but this has yet to be proven.

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Applications

 # The Fibonacci numbers are important in the run-time analysis of Euclid’s Algorithm to determine the greatest         common divisor of two integers.

# The Fibonacci numbers occur in a formula about the diagonals of Pascal's triangle.

# Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation.

# Fibonacci numbers are also used by some pseudorandom number generators.

# In music, Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements.