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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.

Contents

Recurrence relationsEdit

Given two integer parameters P and Q, the Lucas sequences of the first kind Un(P,Q) and of the second kind Vn(P,Q) are defined by the recurrence relations:

 

and

 

It is not hard to show that for  ,

 

ExamplesEdit

Initial terms of Lucas sequences Un(P,Q) and Vn(P,Q) are given in the table:

 

Explicit expressionsEdit

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 rootsEdit

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 rootEdit

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

 
 .

PropertiesEdit

Generating functionsEdit

The ordinary generating functions are

 
 

Sequences with the same discriminantEdit

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

 
 

have the same discriminant:  .

Pell equationsEdit

When  , the Lucas sequences   and   satisfy certain Pell equations:

 
 
 

Other relationsEdit

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:[1]

  • If n / m is odd, 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 even and Q is odd, then the parity of   is the same as n and   is always even.
  • If P is odd and Q is even, then   are always odd for  .
  • 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 and divides P and Q, then p divides   for every  .
  • If p is an odd prime and divides P but not Q, then p divides   if and only if n is even.
  • If p is an odd prime and divides not P but Q, then p never divides   for  .
  • If p is an odd prime and divides not PQ but D, then p divides   if and only if p divides n.
  • If p is an odd prime and 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. 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.[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 namesEdit

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 (Yabuta 2001).
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 base x
Vn(x+1, x) : xn + 1

Some Lucas sequences have entries in the On-Line Encyclopedia of Integer Sequences:

       
−1 3 OEISA214733
1 −1 OEISA000045 OEISA000032
1 1 OEISA128834 OEISA087204
1 2 OEISA107920 OEISA002249
2 −1 OEISA000129 OEISA002203
2 1 OEISA001477
2 2 OEISA009545 OEISA007395
2 3 OEISA088137
2 4 OEISA088138
2 5 OEISA045873
3 −5 OEISA015523 OEISA072263
3 −4 OEISA015521 OEISA201455
3 −3 OEISA030195 OEISA172012
3 −2 OEISA007482 OEISA206776
3 −1 OEISA006190 OEISA006497
3 1 OEISA001906 OEISA005248
3 2 OEISA000225 OEISA000051
3 5 OEISA190959
4 −3 OEISA015530 OEISA080042
4 −2 OEISA090017
4 −1 OEISA001076 OEISA014448
4 1 OEISA001353 OEISA003500
4 2 OEISA007070 OEISA056236
4 3 OEISA003462 OEISA034472
4 4 OEISA001787
5 −3 OEISA015536
5 −2 OEISA015535
5 −1 OEISA052918 OEISA087130
5 1 OEISA004254 OEISA003501
5 4 OEISA002450 OEISA052539
6 1 OEISA001109 OEISA003499

ApplicationsEdit

  • 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.

See alsoEdit

NotesEdit

  1. ^ For such relations and divisibility properties, see (Carmichael 1913), (Lehmer 1930) or (Ribenboim 1996, 2.IV).
  2. ^ Yabuta, M (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly. 39: 439–443. Retrieved 4 October 2018.
  3. ^ Bilu, Yuri; Hanrot, Guillaume; Voutier, Paul M.; Mignotte, Maurice (2001). "Existence of primitive divisors of Lucas and Lehmer numbers". J. Reine Angew. Math. 539: 75–122. doi:10.1515/crll.2001.080. MR 1863855.
  4. ^ John Brillhart; Derrick Henry Lehmer; John Selfridge (April 1975). "New Primality Criteria and Factorizations of 2m ± 1". Mathematics of Computation. 29 (130): 620–647. doi:10.1090/S0025-5718-1975-0384673-1. JSTOR 2005583.
  5. ^ P. J. Smith; M. J. J. Lennon (1993). "LUC: A new public key system". Proceedings of the Ninth IFIP Int. Symp. on Computer Security: 103–117.
  6. ^ D. Bleichenbacher; W. Bosma; A. K. Lenstra (1995). "Some Remarks on Lucas-Based Cryptosystems" (PDF). Lecture Notes in Computer Science. 963: 386–396. doi:10.1007/3-540-44750-4_31.

ReferencesEdit