Other Relations
The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers and Lucas numbers . For example:
| General | P = 1, Q = -1 |
|---|---|
Among the consequences is that is a multiple of, i.e., the sequence is a divisibility sequence. This implies, in particular, that can be prime only when n is prime. Another consequence is an analog of exponentiation by squaring that allows fast computation of for large values of n. These facts are used in the Lucas–Lehmer primality test.
Carmichael's theorem states that all but finitely many of the terms in a Lucas sequence have a prime factor that does not divide any earlier term in the sequence (Yubuta 2001).
Read more about this topic: Lucas Sequence
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