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Lucas sequence

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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 (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:


It is not hard to show that for ,

The above relations can be stated in matrix form as follows:


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:


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


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


  • 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.


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

See also[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
  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.