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.
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:
It is not hard to show that for ,
The ordinary generating functions are
Initial terms of Lucas sequences Un(P,Q) and Vn(P,Q) are given in the table:
The characteristic equation of the recurrence relation for Lucas sequences and is:
It has the discriminant and the roots:
Note that the sequence and the sequence also satisfy the recurrence relation. However these might not be integer sequences.
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
The case occurs exactly when for some integer S so that . In this case one easily finds that
Additional sequences having the same discriminant
If the Lucas sequences and have discriminant , then the sequences based on and where
have the same discriminant: .
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. These facts are used in the Lucas–Lehmer primality test.
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) : Companion Pell numbers or Pell-Lucas 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 (Yubuta 2001).
- Un(x,−1) : Fibonacci polynomials
- Vn(x,−1) : Lucas polynomials
- 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 A214733 1 -1 A000045 A000032 1 1 A128834 A087204 1 2 A107920 2 -1 A000129 A002203 2 1 A001477 2 2 A009545 A007395 2 3 A088137 2 4 A088138 2 5 A045873 3 -5 A015523 A072263 3 -4 A015521 A201455 3 -3 A030195 A172012 3 -2 A007482 A206776 3 -1 A006190 A006497 3 1 A001906 A005248 3 2 A000225 A000051 3 5 A190959 4 -3 A015530 A080042 4 -2 A090017 4 -1 A001076 A014448 4 1 A001353 A003500 4 2 A007070 A056236 4 3 A003462 A034472 4 4 A001787 5 -3 A015536 5 -2 A015535 5 -1 A052918 A087130 5 1 A004254 A003501 5 4 A002450 A052539
- 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
- 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, a paper by Bleichenbacher et al. 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.
- John Brillhart; Derrick Henry Lehmer; John Selfridge (April 1975). "New Primality Criteria and Factorizations of 2^m ± 1". Mathematics of Computation. 29 (130): 620–647. doi:10.1090/S0025-5718-1975-0384673-1.
- 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.
- 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.
- 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. 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. doi:10.2140/pjm.1985.118.449. MR 789184.
- Hans Riesel (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. 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. Numb. Theory. 58 (1): 104–123. doi:10.1006/jnth.1996.0068.
- Joye, M.; Quisquater, J.-J. (1996). "Efficient computation of full Lucas sequences" (PDF). El. Lett. 32 (6): 537–538. doi:10.1049/el:19960359.
- 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.
- Yabuta, M. (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly. 39: 439–443..
- Arthur T. Benjamin; Jennifer J. Quinn (2003). Proofs that Really Count. Mathematical Association of America. p. 35. ISBN 0-88385-333-7.
- Lucas sequence at Encyclopedia of Mathematics.
- Weisstein, Eric W. "Lucas Sequence". MathWorld.