imported>Hendra I. Nurdin |
imported>Hendra I. Nurdin |
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| *<math>\scriptstyle V_{2n} = V_n^2 - 2Q^n\ </math> | | *<math>\scriptstyle V_{2n} = V_n^2 - 2Q^n\ </math> |
| *<math>\scriptstyle \operatorname{ggT}(U_m,U_n)=U_{\operatorname{ggT}(m,n)}</math> | | *<math>\scriptstyle \operatorname{ggT}(U_m,U_n)=U_{\operatorname{ggT}(m,n)}</math> |
| *<math>\scriptstyle m\mid n\implies U_m\mid U_n</math> for all <math>U_m\ne 1</math> | | *<math>\scriptstyle m\mid n\implies U_m\mid U_n</math> for all <math>\scriptstyle U_m\ne 1</math> |
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| ==Fibonacci numbers and Lucas numbers== | | ==Fibonacci numbers and Lucas numbers== |
Revision as of 02:39, 17 November 2007
Lucas sequences are a particular generalisation of sequences like the Fibonacci numbers, Lucas numbers, Pell numbers or Jacobsthal numbers. These sequences have one common characteristic: they can be generated over quadratic equations of the form:
.
There exists 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