imported>Wlodzimierz Holsztynski |
imported>Wlodzimierz Holsztynski |
Line 91: |
Line 91: |
|
| |
|
| :::<math>\Phi\ =\ \lim_{n\to\infty}\frac{F(n+1)}{F(n)}\ =\ \frac{1+\sqrt{5}}{2}</math> | | :::<math>\Phi\ =\ \lim_{n\to\infty}\frac{F(n+1)}{F(n)}\ =\ \frac{1+\sqrt{5}}{2}</math> |
| | |
|
| |
|
| == Further reading == | | == Further reading == |
| * [[John Horton Conway|John H. Conway]] und Richard K. Guy, ''The Book of Numbers'', ISBN 0-387-97993-X | | * [[John Horton Conway|John H. Conway]] und Richard K. Guy, ''The Book of Numbers'', ISBN 0-387-97993-X |
Revision as of 21:56, 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.)

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