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 above relations can be stated in matrix form as follows:
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
The ordinary generating functions are
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
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:
- 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 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. 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 shows that if n > 30, then has a primitive prime factor and determines all cases has no primitive prime factor.
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
- 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 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 2 2 OEIS: A009545 OEIS: A007395 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
- 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.
- For such relations and divisibility properties, see (Carmichael 1913), (Lehmer 1930) or (Ribenboim 1996, 2.IV).
- Yabuta, M (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly. 39: 439–443. Retrieved 4 October 2018.
- 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.
- 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.
- 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.
- 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. ISBN 978-3-540-60221-7.
- 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.
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- 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. 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. 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.
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- Benjamin, Arthur T.; Quinn, Jennifer J. (2003). Proofs that Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. 27. Mathematical Association of America. p. 35. ISBN 978-0-88385-333-7.
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