Wikipedia / Lucas sequence
Wikipedia / Lucas sequence
In mathematics, the Lucas sequences are certain constant-recursive integer sequences that satisfy the recurrence relation
are fixed integers satisfy the same recurrence relation, they differ on the values of their first two elements and thus differ for subsequent elements as well. Any sequence satisfying this recurrence relation can be represented as a linear combination
represent sequences of polynomials 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 (see below). Lucas sequences are named after the French mathematician Édouard Lucas.
Recurrence relations
are defined by the recurrence relations:
and
,
The above relations can be stated in matrix form as follows:
are given in the table:
Explicit expressions
is:
It has the discriminant and, by the quadratic formula, has the roots:
Thus:
also satisfy the recurrence relation. However these might not be integer sequences.
Distinct roots
, 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
for some integer S. In this case one easily finds that
Properties
Generating functions
The ordinary generating functions are
Pell equations
satisfy certain Pell equations:
Relations between sequences with different parameters
For any number c with
:
For any number c, we also have
Other relations
The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers and Lucas numbers. For example:
Of these, (6) and (7) allow fast calculation of V independent of U in a way analogous to exponentiation by squaring (which belongs to the section above, "relations between sequences with different parameters") is also useful for this purpose.
Fast computation
An analog of exponentiation by squaring allows for large values of n.
Divisibility properties
is a divisibility sequence can be prime only when n is a strong divisibility sequence.
Other divisibility properties are as follows:
If n is an odd multiple of m.
Let N be an integer relatively prime to 2Q. If the smallest positive integer r for which N exists, then the set of n for which N is exactly the set of multiples of r.
If P and Q are even.
If P is odd and Q.
If P is even and Q is odd, then the parity is the same as n is always even.
If P and Q are even if and only if n is a multiple of 3.
If p (see Legendre symbol).
If p is an odd prime which divides P and Q, then p.
If p is an odd prime which divides P but not Q, then p if and only if n is even.
If p is an odd prime which divides Q but not P, then p.
If p is an odd prime which divides D but not PQ, then p if and only if p divides n.
If p is an odd prime which does not divide PQD, then p.
The last fact generalizes Fermat's little theorem. These facts are used in the Lucas–Lehmer primality test. Like Fermat's little theorem, the converse of the last fact holds often, but not always; there exist composite numbers n relatively prime to D. Such composite numbers are called Lucas pseudoprimes.
A prime factor of a term in a Lucas sequence which 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 (1913) showed that if D is positive and n has a primitive prime factor. In the case D is negative, a deep result of Bilu, Hanrot, Voutier and Mignotte shows that if n has no primitive prime factor.
Specific names
The Lucas sequences for some values of and have specific names:
: Pell–Lucas numbers (companion Pell numbers)
: Numbers of the form , which include the Fermat numbers
: The square roots of the square triangular numbers.
: Chebyshev polynomials of second kind
: Chebyshev polynomials of first kind multiplied by 2
: Repunits in base
:
Some Lucas sequences have entries in the On-Line Encyclopedia of Integer Sequences:
{|class="wikitable" style="background: #fff"
|- | −1 || 3 || |- | 1 || −1 || || |- | 1 || 1 || || |- | 1 || 2 || || |- | 2 || −1 || || |- | 2 || 1 || || |- | 2 || 2 || |- | 2 || 3 || |- | 2 || 4 || |- | 2 || 5 || |- | 3 || −5 || || |- | 3 || −4 || || |- | 3 || −3 || || |- | 3 || −2 || || |- | 3 || −1 || || |- | 3 || 1 || || |- | 3 || 2 || || |- | 3 || 5 || |- | 4 || −3 || || |- | 4 || −2 || |- | 4 || −1 || || |- | 4 || 1 || || |- | 4 || 2 || || |- | 4 || 3 || || |- | 4 || 4 || |- | 5 || −3 || |- | 5 || −2 || |- | 5 || −1 || || |- | 5 || 1 || || |- | 5 || 4 || || |- | 6 || 1 || || |}
Applications
Lucas sequences are used in probabilistic Lucas pseudoprime tests, which are part of the commonly used Baillie–PSW primality test.
Lucas sequences are used in some primality proof methods, including the Lucas–Lehmer and Lucas–Lehmer–Riesel tests and the hybrid N−1/N+1 methods such as those in Brillhart-Lehmer-Selfridge 1975.
LUC is a public-key cryptosystem based on Lucas sequences that implements the analogs of ElGamal (LUCELG), Diffie–Hellman (LUCDIF), and RSA (LUCRSA). The encryption of the message in LUC is computed as a term of certain Lucas sequence, instead of using modular exponentiation as in RSA or Diffie–Hellman. However, it is argued that many of the supposed security advantages of LUC over cryptosystems based on modular exponentiation are either not present, or not as substantial as claimed.
Generalizations
when and are the roots of the corresponding quadratic equation , generalizes to degree with integers and typically with , let is a sequence of integers satisfying the recurrence, as is evidenced by its ordinary generating function
Software
SageMath as functions lucas_number1() and lucas_number2(), respectively.
See also
Notes
References