imported>Karsten Meyer |
imported>Hendra I. Nurdin |
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| '''Lucas sequences''' are a particular generalisation of sequences like the [[Fibonacci number|Fibonacci numbers]], [[Lucas number|Lucas numbers]], [[Pell number|Pell numbers]] or [[Jacobsthal number|Jacobsthal numbers]]. These sequences have one common characteristic: they can be generated over [[quadratic equation|quadratic equations]] of the form: <math>\scriptstyle x^2-Px+Q=0\ </math> with <math>\scriptstyle P^2-4Q \ne 0</math>. | | {{subpages}} |
| | In [[mathematics]], a '''Lucas sequence''' is a particular generalisation of sequences like the [[Fibonacci number|Fibonacci numbers]], [[Lucas number|Lucas numbers]], [[Pell number|Pell numbers]] or [[Jacobsthal number|Jacobsthal numbers]]. Lucas sequences have one common characteristic: they can be generated over [[quadratic equation|quadratic equations]] of the form: <math>\scriptstyle x^2-Px+Q=0\ </math> with <math>\scriptstyle P^2-4Q \ne 0</math>. |
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| There exist two kinds of Lucas sequences: | | There exist two kinds of Lucas sequences: |
Revision as of 23:46, 17 November 2007
In mathematics, a Lucas sequence is a particular generalisation of sequences like the Fibonacci numbers, Lucas numbers, Pell numbers or Jacobsthal numbers. Lucas sequences have one common characteristic: they can be generated over quadratic equations of the form:
with
.
There exist two kinds of Lucas sequences:
- Sequences
with
,
- Sequences
with
,
where
and
are the solutions

and

of the quadratic equation
.
Properties
- The variables
and
, and the parameter
and
are interdependent. In particular,
and
.
- For every sequence
it holds that
and
.
- For every sequence
is holds that
and
.
For every Lucas sequence the following are true:




for all 
Fibonacci numbers and Lucas numbers
The two best known Lucas sequences are the Fibonacci numbers
and the Lucas numbers
with
and
.
Lucas sequences and the prime numbers
If the natural number
is a prime number then it holds that
divides 
divides 
Fermat's Little Theorem can then be seen as a special case of
divides
because
is equivalent to
.
The converse pair of statements that if
divides
then is
a prime number and if
divides
then is
a prime number) are individually false and lead to Fibonacci pseudoprimes and Lucas pseudoprimes, respectively.
Further reading