Lucas sequence

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is developing and not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

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


Recurrence relation

The Lucas sequences U(P,Q) and V(P,Q) are defined by the recurrence relations

and


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