Lucas Sequence
المؤلف:
Ribenboim, P.
المصدر:
The Little Book of Big Primes. New York: Springer-Verlag
الجزء والصفحة:
...
1-11-2020
944
Lucas Sequence
Let 
, 
be integers satisfying
 |
(1)
|
Then roots of
 |
(2)
|
are
so
Now define
for integer 
, so the first few values are
and
Closed forms for these are given by
The sequences
are called Lucas sequences, where the definition is usually extended to include
 |
(37)
|
The following table summarizes special cases of 
and 
.
 |
 |
 |
 |
Fibonacci numbers |
Lucas numbers |
 |
Pell numbers |
Pell-Lucas numbers |
 |
Jacobsthal numbers |
Pell-Jacobsthal numbers |
The Lucas sequences satisfy the general recurrence relations
Taking 
then gives
Other identities include
These formulas allow calculations for large 
to be decomposed into a chain in which only four quantities must be kept track of at a time, and the number of steps needed is 
. The chain is particularly simple if 
has many 2s in its factorization.
REFERENCES:
Dickson, L. E. "Recurring Series; Lucas' 
, 
." Ch. 17 in History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 393-411, 2005.
Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, pp. 35-53, 1991.
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