Lucas sequence

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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


of the quadratic equation .


  • 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


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