Lucas sequence
Certain constant-recursive integer sequences From Wikipedia, the free encyclopedia
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 (see below). Lucas sequences are named after the French mathematician Édouard Lucas.
Recurrence relations
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Perspective
Given two integer parameters and , the Lucas sequences of the first kind and of the second kind are defined by the recurrence relations:
and
It is not hard to show that for ,
The above relations can be stated in matrix form as follows:
Examples
Initial terms of Lucas sequences and are given in the table:
Explicit expressions
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Perspective
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
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Generating functions
The ordinary generating functions are
Pell equations
When , the Lucas sequences and satisfy certain Pell equations:
Relations between sequences with different parameters
- For any number c, the sequences and with
- have the same discriminant as and :
- 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:
Divisibility properties
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:[1]
- If n is an odd multiple of m, then divides .
- Let N be an integer relatively prime to 2Q. If the smallest positive integer r for which N divides exists, then the set of n for which N divides is exactly the set of multiples of r.
- If P and Q are even, then are always even except .
- If P is odd and Q is even, then are always odd for every .
- If P is even and Q is odd, then the parity of is the same as n and is always even.
- If P and Q are odd, then are even if and only if n is a multiple of 3.
- If p is an odd prime, then (see Legendre symbol).
- If p is an odd prime which divides P and Q, then p divides for every .
- If p is an odd prime which divides P but not Q, then p divides if and only if n is even.
- If p is an odd prime which divides Q but not P, then p never divides for any .
- If p is an odd prime which divides D but not PQ, then p divides if and only if p divides n.
- If p is an odd prime which does not divide PQD, then p divides , where .
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 and dividing , where . 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.[2] Indeed, Carmichael (1913) 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[3] shows that if n > 30, then has a primitive prime factor and determines all cases has no primitive prime factor.
Specific names
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The Lucas sequences for some values of P and Q have specific names:
- Un(1, −1) : Fibonacci numbers
- Vn(1, −1) : Lucas numbers
- Un(2, −1) : Pell numbers
- Vn(2, −1) : Pell–Lucas numbers (companion Pell numbers)
- Un(1, −2) : Jacobsthal numbers
- Vn(1, −2) : Jacobsthal–Lucas numbers
- Un(3, 2) : Mersenne numbers 2n − 1
- Vn(3, 2) : Numbers of the form 2n + 1, which include the Fermat numbers[2]
- Un(6, 1) : The square roots of the square triangular numbers.
- Un(x, −1) : Fibonacci polynomials
- Vn(x, −1) : Lucas polynomials
- Un(2x, 1) : Chebyshev polynomials of second kind
- Vn(2x, 1) : Chebyshev polynomials of first kind multiplied by 2
- Un(x+1, x) : Repunits in base x
- Vn(x+1, x) : xn + 1
Some Lucas sequences have entries in the On-Line Encyclopedia of Integer Sequences:
−1 | 3 | OEIS: A214733 | |
1 | −1 | OEIS: A000045 | OEIS: A000032 |
1 | 1 | OEIS: A128834 | OEIS: A087204 |
1 | 2 | OEIS: A107920 | OEIS: A002249 |
2 | −1 | OEIS: A000129 | OEIS: A002203 |
2 | 1 | OEIS: A001477 | OEIS: A007395 |
2 | 2 | OEIS: A009545 | |
2 | 3 | OEIS: A088137 | |
2 | 4 | OEIS: A088138 | |
2 | 5 | OEIS: A045873 | |
3 | −5 | OEIS: A015523 | OEIS: A072263 |
3 | −4 | OEIS: A015521 | OEIS: A201455 |
3 | −3 | OEIS: A030195 | OEIS: A172012 |
3 | −2 | OEIS: A007482 | OEIS: A206776 |
3 | −1 | OEIS: A006190 | OEIS: A006497 |
3 | 1 | OEIS: A001906 | OEIS: A005248 |
3 | 2 | OEIS: A000225 | OEIS: A000051 |
3 | 5 | OEIS: A190959 | |
4 | −3 | OEIS: A015530 | OEIS: A080042 |
4 | −2 | OEIS: A090017 | |
4 | −1 | OEIS: A001076 | OEIS: A014448 |
4 | 1 | OEIS: A001353 | OEIS: A003500 |
4 | 2 | OEIS: A007070 | OEIS: A056236 |
4 | 3 | OEIS: A003462 | OEIS: A034472 |
4 | 4 | OEIS: A001787 | |
5 | −3 | OEIS: A015536 | |
5 | −2 | OEIS: A015535 | |
5 | −1 | OEIS: A052918 | OEIS: A087130 |
5 | 1 | OEIS: A004254 | OEIS: A003501 |
5 | 4 | OEIS: A002450 | OEIS: A052539 |
6 | 1 | OEIS: A001109 | OEIS: A003499 |
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–Riesel test, and the N+1 and hybrid N−1/N+1 methods such as those in Brillhart-Lehmer-Selfridge 1975.[4]
- LUC is a public-key cryptosystem based on Lucas sequences[5] 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, a paper by Bleichenbacher et al.[6] shows 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.
Software
Sagemath implements and as lucas_number1()
and lucas_number2()
, respectively.[7]
See also
Notes
References
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