# Lucas sequence

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 .

## Contents

## 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

- P. Ribenboim,
*The New Book of Prime Number Records*(3 ed.), Springer, 1996, ISBN 0-387-94457-5. - P. Ribenboim,
*My Numbers, My Friends*, Springer, 2000, ISBN 0-387-98911-0.