Lucas sequence


In mathematics, the Lucas sequences and are certain constant-recursive integer sequences that satisfy the recurrence relation
where and are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences and.
More generally, Lucas sequences and represent sequences of polynomials in and with integer coefficients.
Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers. Lucas sequences are named after the French mathematician Édouard Lucas.

Recurrence relations

Given two integer parameters P and Q, the Lucas sequences of the first kind Un and of the second kind Vn are defined by the recurrence relations:
and
It is not hard to show that for,

Examples

Initial terms of Lucas sequences Un and Vn are given in the table:

Explicit expressions

The characteristic equation of the recurrence relation for Lucas sequences and is:
It has the discriminant and the roots:
Thus:
Note that the sequence and the sequence also satisfy the recurrence relation. However these might not be integer sequences.

Distinct roots

When, a and b are distinct and one quickly verifies that
It follows that the terms of Lucas sequences can be expressed in terms of a and b as follows

Repeated root

The case occurs exactly when for some integer S so that. In this case one easily finds that

Properties

Generating functions

The ordinary generating functions are

Sequences with the same discriminant

If the Lucas sequences and have
discriminant, then the sequences based on and where
have the same discriminant:.

Pell equations

When, the Lucas sequences and satisfy certain Pell equations:

Other relations

The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers and Lucas numbers. For example:
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.
Moreover, if, then is a strong divisibility sequence.
Other divisibility properties are as follows:
The last fact generalizes Fermat's little theorem. These facts are used in the Lucas–Lehmer primality test.
The converse of the last fact does not hold, as the converse of Fermat's little theorem does not hold. There exists a composite n relatively prime to D and dividing, where. Such a composite is called Lucas pseudoprime.
A prime factor of a term in a Lucas sequence that does not divide any earlier term in the sequence is called primitive.
Carmichael's theorem states that all but finitely many of the terms in a Lucas sequence have a primitive prime factor. Indeed, Carmichael showed that if D is positive and n is not 1, 2 or 6, then has a primitive prime factor. In the case D is negative, a deep result of Bilu, Hanrot, Voutier and Mignotte shows that if n > 30, then has a primitive prime factor and determines all cases has no primitive prime factor.

Specific names

The Lucas sequences for some values of P and Q have specific names:
Some Lucas sequences have entries in the On-Line Encyclopedia of Integer Sequences:

Applications