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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.