imported>Wlodzimierz Holsztynski |
imported>Wlodzimierz Holsztynski |
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| ==Properties== | | ==Properties== |
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| Below, we will apply the following simple observation to Fibonacci numbers:
| | We will apply the following simple observation to Fibonacci numbers: |
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| if three integers <math>\ a,b,c,</math> satisfy equality <math>\ c = a+b,</math> then | | if three integers <math>\ a,b,c,</math> satisfy equality <math>\ c = a+b,</math> then |
Revision as of 21:47, 29 December 2007
In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation:

The sequence of fibonacci numbers start: 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 then
is prime. (The converse is false.)

Direct formula
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
. It follows that
; thus the value of the golden ratio is
.
Further reading