Revision as of 22:39, 15 November 2007 by imported>Karsten Meyer
Lucas sequences are the particular generalisation of sequences like Fibonacci numbers, Lucas numbers, Pell numbers or Jacobsthal numbers. Every of this sequences has one common factor. They could be generatet over quadratic equatations of the form:
.
There exists kinds of Lucas sequences:
- Sequence
with 
- Sequence
with 
and
are the solutions
and
of the quadratic equatation
.
Properties
- The variables
and
, and the parameter
and
are interdependent. So it is true, that
and
.
- For every sequence
is it true, that
and
.
- For every sequence
is it true, that
and
.
For every Lucas sequence is true that




; für alle 
Fibonacci numbers and Lucas numbers
The both best-known Lucas sequences are the Fibonacci numbers
and the Lucas numbers
with
and
.
Lucas sequences and the Prime numbers
Is the natural number
a Prime number, then it is true, that
divides 
divides 
Fermat's little theorem you can see as a special case of
divides
because
is äquivalent to
The converse (If
divides
then is
a prime number and if
divides
then is
a prime number) is false and lead to Fibonacci pseudoprimes respectively to Lucas pseudoprimes.
Further reading