Lucas sequence

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Revision as of 02:41, 17 November 2007 by imported>Hendra I. Nurdin
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Lucas sequences are a particular generalisation of sequences like the Fibonacci numbers, Lucas numbers, Pell numbers or Jacobsthal numbers. These sequences have one common characteristic: they can be generated over quadratic equations of the form: .

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

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