Lucas sequence

Summary

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 edit

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 edit

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

 

Explicit expressions edit

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 edit

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 edit

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

 
 

Properties edit

Generating functions edit

The ordinary generating functions are

 
 

Pell equations edit

When  , the Lucas sequences   and   satisfy certain Pell equations:

 
 
 

Relations between sequences with different parameters edit

  • For any number c, the sequences   and   with
 
 
have the same discriminant as   and  :
 
  • For any number c, we also have
 
 

Other relations edit

The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers   and Lucas numbers  . For example:

 

Divisibility properties edit

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   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 a 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 names edit

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 OEISA214733
1 −1 OEISA000045 OEISA000032
1 1 OEISA128834 OEISA087204
1 2 OEISA107920 OEISA002249
2 −1 OEISA000129 OEISA002203
2 1 OEISA001477 OEISA007395
2 2 OEISA009545
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

Applications edit

  • 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 edit

Sagemath implements   and   as lucas_number1() and lucas_number2(), respectively.[7]

See also edit

Notes edit

  1. ^ For such relations and divisibility properties, see (Carmichael 1913), (Lehmer 1930) or (Ribenboim 1996, 2.IV).
  2. ^ a b 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" (PDF). J. Reine Angew. Math. 2001 (539): 75–122. doi:10.1515/crll.2001.080. MR 1863855. S2CID 122969549.
  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. CiteSeerX 10.1.1.32.1835.
  6. ^ D. Bleichenbacher; W. Bosma; A. K. Lenstra (1995). "Some Remarks on Lucas-Based Cryptosystems" (PDF). Advances in Cryptology — CRYPT0' 95. Lecture Notes in Computer Science. Vol. 963. pp. 386–396. doi:10.1007/3-540-44750-4_31. ISBN 978-3-540-60221-7.
  7. ^ "Combinatorial Functions - Combinatorics". doc.sagemath.org. Retrieved 2023-07-13.

References edit

  • Carmichael, R. D. (1913), "On the numerical factors of the arithmetic forms αn±βn", Annals of Mathematics, 15 (1/4): 30–70, doi:10.2307/1967797, JSTOR 1967797
  • Lehmer, D. H. (1930). "An extended theory of Lucas' functions". Annals of Mathematics. 31 (3): 419–448. Bibcode:1930AnMat..31..419L. doi:10.2307/1968235. JSTOR 1968235.
  • Ward, Morgan (1954). "Prime divisors of second order recurring sequences". Duke Math. J. 21 (4): 607–614. doi:10.1215/S0012-7094-54-02163-8. hdl:10338.dmlcz/137477. MR 0064073.
  • Somer, Lawrence (1980). "The divisibility properties of primary Lucas Recurrences with respect to primes" (PDF). Fibonacci Quarterly. 18: 316.
  • Lagarias, J. C. (1985). "The set of primes dividing Lucas Numbers has density 2/3". Pac. J. Math. 118 (2): 449–461. CiteSeerX 10.1.1.174.660. doi:10.2140/pjm.1985.118.449. MR 0789184.
  • Hans Riesel (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Vol. 126 (2nd ed.). Birkhäuser. pp. 107–121. ISBN 0-8176-3743-5.
  • Ribenboim, Paulo; McDaniel, Wayne L. (1996). "The square terms in Lucas Sequences". J. Number Theory. 58 (1): 104–123. doi:10.1006/jnth.1996.0068.
  • Joye, M.; Quisquater, J.-J. (1996). "Efficient computation of full Lucas sequences" (PDF). Electronics Letters. 32 (6): 537–538. Bibcode:1996ElL....32..537J. doi:10.1049/el:19960359. Archived from the original (PDF) on 2015-02-02.
  • Ribenboim, Paulo (1996). The New Book of Prime Number Records (eBook ed.). Springer-Verlag, New York. doi:10.1007/978-1-4612-0759-7. ISBN 978-1-4612-0759-7.
  • Ribenboim, Paulo (2000). My Numbers, My Friends: Popular Lectures on Number Theory. New York: Springer-Verlag. pp. 1–50. ISBN 0-387-98911-0.
  • Luca, Florian (2000). "Perfect Fibonacci and Lucas numbers". Rend. Circ Matem. Palermo. 49 (2): 313–318. doi:10.1007/BF02904236. S2CID 121789033.
  • Yabuta, M. (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly. 39: 439–443.
  • Benjamin, Arthur T.; Quinn, Jennifer J. (2003). Proofs that Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. Vol. 27. Mathematical Association of America. p. 35. ISBN 978-0-88385-333-7.
  • Lucas sequence at Encyclopedia of Mathematics.
  • Weisstein, Eric W. "Lucas Sequence". MathWorld.
  • Wei Dai. "Lucas Sequences in Cryptography".