# Lucas sequence

Main Article
Talk
Related Articles  [?]
Bibliography  [?]
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.