Lucas sequence

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Certain constant-recursive integer sequences

title: "Lucas sequence" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["recurrence-relations", "integer-sequences"] description: "Certain constant-recursive integer sequences" topic_path: "general/recurrence-relations" source: "https://en.wikipedia.org/wiki/Lucas_sequence" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Certain constant-recursive integer sequences ::

In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation

: x_n = P \cdot x_{n - 1} - Q \cdot x_{n - 2}

where P and Q are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U_n(P, Q) and V_n(P, Q).

More generally, Lucas sequences U_n(P, Q) and V_n(P, Q) represent sequences of polynomials in P and Q 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

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

:\begin{align} U_0(P,Q)&=0, \ U_1(P,Q)&=1, \ U_n(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q) \mbox{ for }n1, \end{align}

and

:\begin{align} V_0(P,Q)&=2, \ V_1(P,Q)&=P, \ V_n(P,Q)&=P\cdot V_{n-1}(P,Q)-Q\cdot V_{n-2}(P,Q) \mbox{ for }n1. \end{align}

It is not hard to show that for n0,

:\begin{align} U_n(P,Q)&=\frac{P\cdot U_{n-1}(P,Q) + V_{n-1}(P,Q)}{2}, \ V_n(P,Q)&=\frac{(P^2-4Q)\cdot U_{n-1}(P,Q)+P\cdot V_{n-1}(P,Q)}{2}. \end{align}

The above relations can be stated in matrix form as follows: : \begin{bmatrix} U_n(P,Q)\ U_{n+1}(P,Q)\end{bmatrix} = \begin{bmatrix} 0 & 1\ -Q & P\end{bmatrix}\cdot \begin{bmatrix} U_{n-1}(P,Q)\ U_n(P,Q)\end{bmatrix}, : \begin{bmatrix} V_n(P,Q)\ V_{n+1}(P,Q)\end{bmatrix} = \begin{bmatrix} 0 & 1\ -Q & P\end{bmatrix}\cdot \begin{bmatrix} V_{n-1}(P,Q)\ V_n(P,Q)\end{bmatrix}, : \begin{bmatrix} U_n(P,Q)\ V_n(P,Q)\end{bmatrix} = \begin{bmatrix} P/2 & 1/2\ (P^2-4Q)/2 & P/2\end{bmatrix}\cdot \begin{bmatrix} U_{n-1}(P,Q)\ V_{n-1}(P,Q)\end{bmatrix}.

Initial terms of Lucas sequences U_n(P,Q) and V_n(P,Q) are given in the table: : \begin{array}{r|l|l} n & U_n(P,Q) & V_n(P,Q) \ \hline 0 & 0 & 2 \ 1 & 1 & P \ 2 & P & {P}^{2}-2Q \ 3 & {P}^{2}-Q & {P}^{3}-3PQ \ 4 & {P}^{3}-2PQ & {P}^{4}-4{P}^{2}Q+2{Q}^{2} \ 5 & {P}^{4}-3{P}^{2}Q+{Q}^{2} & {P}^{5}-5{P}^{3}Q+5P{Q}^{2} \ 6 & {P}^{5}-4{P}^{3}Q+3P{Q}^{2} & {P}^{6}-6{P}^{4}Q+9{P}^{2}{Q}^{2}-2{Q}^{3} \end{array}

Explicit expressions

The characteristic equation of the recurrence relation for Lucas sequences U_n(P,Q) and V_n(P,Q) is: :x^2 - Px + Q=0 , It has the discriminant D = P^2 - 4Q and, by the quadratic formula, has the roots: :a = \frac{P+\sqrt{D}}2\quad\text{and}\quad b = \frac{P-\sqrt{D}}2. ,

Thus: :a + b = P, , :a b = \frac{1}{4}(P^2 - D) = Q, , :a - b = \sqrt{D}, .

Note that the sequence a^n and the sequence b^n also satisfy the recurrence relation. However these might not be integer sequences.

Distinct roots

When D\ne 0, a and b are distinct and one quickly verifies that

:a^n = \frac{V_n + U_n \sqrt{D}}{2}

:b^n = \frac{V_n - U_n \sqrt{D}}{2}.

It follows that the terms of Lucas sequences can be expressed in terms of a and b as follows

:U_n = \frac{a^n-b^n}{a-b} = \frac{a^n-b^n}{ \sqrt{D}}

:V_n = a^n+b^n ,

Repeated root

The case D=0 occurs exactly when P=2S \text{ and }Q=S^2 for some integer S so that a=b=S. In this case one easily finds that

:U_n(P,Q)=U_n(2S,S^2) = nS^{n-1},

:V_n(P,Q)=V_n(2S,S^2)=2S^n.,

Properties

Generating functions

The ordinary generating functions are : \sum_{n\ge 0} U_n(P,Q)z^n = \frac{z}{1-Pz+Qz^2}; : \sum_{n\ge 0} V_n(P,Q)z^n = \frac{2-Pz}{1-Pz+Qz^2}.

Pell equations

When Q=\pm 1, the Lucas sequences U_n(P, Q) and V_n(P, Q) satisfy certain Pell equations: :V_n(P,1)^2 - D\cdot U_n(P,1)^2 = 4, :V_n(P,-1)^2 - D\cdot U_n(P,-1)^2 = 4(-1)^n.

Relations between sequences with different parameters

  • For any number c, the sequences U_n(P', Q') and V_n(P', Q') with :: P' = P + 2c
    :: Q' = cP + Q + c^2
    :have the same discriminant as U_n(P, Q) and V_n(P, Q): :: P'^2 - 4Q' = (P+2c)^2 - 4(cP + Q + c^2) = P^2 - 4Q = D.

  • For any number c, we also have :: U_n(cP,c^2Q) = c^{n-1}\cdot U_n(P,Q), :: V_n(cP,c^2Q) = c^n\cdot V_n(P,Q).

Other relations

The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers F_n=U_n(1,-1) and Lucas numbers L_n=V_n(1,-1). For example: : \begin{array}{l|l|r} \text{General case} & (P,Q) = (1,-1), D = P^2 - 4Q = 5 \ \hline D U_n = {V_{n+1} - Q V_{n-1}}=2V_{n+1}-P V_n & 5F_n = {L_{n+1} + L_{n-1}}=2L_{n+1} - L_{n} & (1) \ V_n = U_{n+1} - Q U_{n-1}=2U_{n+1}-PU_n & L_n = F_{n+1} + F_{n-1}=2F_{n+1}-F_n & (2) \ U_{m+n} = U_n U_{m+1} - Q U_m U_{n-1} = U_mV_n-Q^nU_{m-n} & F_{m+n} = F_n F_{m+1} + F_m F_{n-1} =F_mL_n-(-1)^nF_{m-n} & (3) \ U_{2n} = U_n (U_{n+1} - QU_{n-1}) = U_n V_n & F_{2n} = F_n (F_{n+1} + F_{n-1}) = F_n L_n & (4) \ U_{2n+1} = U_{n+1}^2 - Q U_n^2 & F_{2n+1} = F_{n+1}^2 + F_n^2 & (5) \ V_{m+n} = V_m V_n - Q^n V_{m-n} = D U_m U_n + Q^n V_{m-n} & L_{m+n} = L_m L_n - (-1)^n L_{m-n} = 5 F_m F_n + (-1)^n L_{m-n} & (6) \ V_{2n} = V_n^2 - 2Q^n = D U_n^2 + 2Q^n & L_{2n} = L_n^2 - 2(-1)^n = 5 F_n^2 + 2(-1)^n & (7) \ U_{m+n} = \frac{U_mV_n+U_nV_m}{2} & F_{m+n} = \frac{F_mL_n+F_nL_m}{2} & (8) \ V_{m+n}=\frac{V_mV_n+DU_mU_n}{2} & L_{m+n}=\frac{L_mL_n+5F_mF_n}{2} & (9) \ V_n^2-DU_n^2=4Q^n & L_n^2-5F_n^2=4(-1)^n & (10) \ U_n^2-U_{n-1}U_{n+1}=Q^{n-1} & F_n^2-F_{n-1}F_{n+1}=(-1)^{n-1} & (11) \ V_n^2-V_{n-1}V_{n+1}=DQ^{n-1} & L_n^2-L_{n-1}L_{n+1}=5(-1)^{n-1} & (12) \ 2^{n-1}U_n={n \choose 1}P^{n-1}+{n \choose 3}P^{n-3}D+\cdots & 2^{n-1}F_n={n \choose 1}+5{n \choose 3}+\cdots & (13) \ 2^{n-1}V_n=P^n+{n \choose 2}P^{n-2}D+{n \choose 4}P^{n-4}D^2+\cdots & 2^{n-1}L_n=1+5{n \choose 2}+5^2{n \choose 4}+\cdots & (14) \end{array}

Of these, (6) and (7) allow fast calculation of V independent of U in a way analogous to exponentiation by squaring. The relation V_{mn} = V_{m}(P = V_n, Q = Q_n) (which belongs to the section above, "relations between sequences with different parameters") is also useful for this purpose.

Fast computation

An analog of exponentiation by squaring applied to the matrix that calculates U_n(P,Q) and V_n(P,Q) from U_{n-1}(P,Q) and V_{n-1}(P,Q) allows -time computation of U_n(P,Q) and V_n(P,Q) for large values of n.

Divisibility properties

Among the consequences is that U_{km}(P,Q) is a multiple of U_m(P,Q), i.e., the sequence (U_m(P,Q)){m\ge1} is a divisibility sequence. This implies, in particular, that U_n(P,Q) can be prime only when n is prime. Moreover, if \gcd(P,Q)=1, then (U_m(P,Q)){m\ge1} is a strong divisibility sequence.

Other divisibility properties are as follows:

  • If n is an odd multiple of m, then V_m divides V_n.
  • Let N be an integer relatively prime to 2Q. If the smallest positive integer r for which N divides U_r exists, then the set of n for which N divides U_n is exactly the set of multiples of r.
  • If P and Q are even, then U_n, V_n are always even except U_1.
  • If P is odd and Q is even, then U_n, V_n are always odd for every n 0.
  • If P is even and Q is odd, then the parity of U_n is the same as n and V_n is always even.
  • If P and Q are odd, then U_n, V_n are even if and only if n is a multiple of 3.
  • If p is an odd prime, then U_p\equiv\left(\tfrac{D}{p}\right), V_p\equiv P\pmod{p} (see Legendre symbol).
  • If p is an odd prime which divides P and Q, then p divides U_n for every n1.
  • If p is an odd prime which divides P but not Q, then p divides U_n if and only if n is even.
  • If p is an odd prime which divides Q but not P, then p never divides U_n for any n 0.
  • If p is an odd prime which divides D but not PQ, then p divides U_n if and only if p divides n.
  • If p is an odd prime which does not divide PQD, then p divides U_l, where l=p-\left(\tfrac{D}{p}\right).

The last fact generalizes Fermat's little theorem. These facts are used in the Lucas–Lehmer primality test. Like Fermat's little theorem, the converse of the last fact holds often, but not always; there exist composite numbers n relatively prime to D and dividing U_l, where l=n-\left(\tfrac{D}{n}\right). Such composite numbers are called Lucas pseudoprimes.

A prime factor of a term in a Lucas sequence which 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 U_n 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 U_n has a primitive prime factor and determines all cases U_n has no primitive prime factor.

Specific names

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(2, 1) : Counting 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 in base x :Vn(x + 1, x) : x**n + 1

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

:{|class="wikitable" style="background: #fff" |- !P,!!Q, !!U_n(P,Q), !! V_n(P,Q), |- | −1 || 3 || |- | 1 || −1 || || |- | 1 || 1 || || |- | 1 || 2 || || |- | 2 || −1 || || |- | 2 || 1 || || |- | 2 || 2 || |- | 2 || 3 || |- | 2 || 4 || |- | 2 || 5 || |- | 3 || −5 || || |- | 3 || −4 || || |- | 3 || −3 || || |- | 3 || −2 || || |- | 3 || −1 || || |- | 3 || 1 || || |- | 3 || 2 || || |- | 3 || 5 || |- | 4 || −3 || || |- | 4 || −2 || |- | 4 || −1 || || |- | 4 || 1 || || |- | 4 || 2 || || |- | 4 || 3 || || |- | 4 || 4 || |- | 5 || −3 || |- | 5 || −2 || |- | 5 || −1 || || |- | 5 || 1 || || |- | 5 || 4 || || |- | 6 || 1 || || |}

Applications

  • 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 and Lucas–Lehmer–Riesel tests and the 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, it is argued 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.

Generalizations

The sequence V_n(P,Q) = a^n + b^n, which is a solution to the recurrence V_n(P,Q) = P V_{n-1}(P,Q) - Q V_{n-2}(P,Q) when and are the roots of the corresponding quadratic equation , generalizes to degree . Specifically, for the recurrence relation V_n(P_1, \ldots, P_k) = \sum_{j=1}^k P_j V_{n-j}(P_1, \ldots, P_k) with integers and typically with , let be the roots of the corresponding polynomial equation z^k - \sum_{j=1}^k P_j z^{k-j} = 0. Then V_n(P_1, \ldots, P_k) = \sum_{j=1}^k a_j^n is a sequence of integers satisfying the recurrence, as is evidenced by its ordinary generating function, G_{P_1, \ldots, P_k}(z) = \sum_{n=0}^\infty V_n(P_1, \ldots, P_k) z^n = \frac{k-\sum_{j=1}^{k-1} (k-j) P_j z^j}{1 - \sum_{j=1}^k P_j z^j}.

Software

  • SageMath implements U_n and V_n as functions lucas_number1() and lucas_number2(), respectively.

Notes

References

  • {{citation | last = Carmichael | first = R. D. | author-link = Robert Daniel Carmichael | doi = 10.2307/1967797 | issue = 1/4 | journal = Annals of Mathematics | pages = 30–70 | title = On the numerical factors of the arithmetic forms αn±βn | volume = 15 | year = 1913 | jstor = 1967797 }}
  • {{cite journal| first1=D. H. | last1=Lehmer |title=An extended theory of Lucas' functions |journal=Annals of Mathematics |year=1930 |volume=31 | number=3 |jstor=1968235 |pages=419–448 |bibcode=1930AnMat..31..419L | doi=10.2307/1968235
  • {{cite journal| first1=Morgan | last1=Ward |title=Prime divisors of second order recurring sequences |journal = Duke Math. J. | year=1954 | volume=21 | number=4 |pages=607–614 | mr=0064073 |doi=10.1215/S0012-7094-54-02163-8 | hdl=10338.dmlcz/137477 | hdl-access=free}}
  • {{cite journal|first1=Lawrence | last1=Somer |title=The divisibility properties of primary Lucas Recurrences with respect to primes |year=1980 | journal=Fibonacci Quarterly | pages=316–334 | volume=18 | issue=4 | doi=10.1080/00150517.1980.12430140 | url=http://www.fq.math.ca/Scanned/18-4/somer.pdf
  • {{cite journal|first1=J. C. | last1=Lagarias |journal=Pac. J. Math. | title=The set of primes dividing Lucas Numbers has density 2/3 |year=1985 | volume=118 | number=2 | pages=449–461 | mr=789184 | doi=10.2140/pjm.1985.118.449 | citeseerx=10.1.1.174.660 }}
  • {{ cite journal|first1=Paulo | last1=Ribenboim | first2=Wayne L. |last2=McDaniel |title=The square terms in Lucas Sequences | journal=J. Number Theory |year=1996 | volume=58 | number=1 | pages=104–123 | doi=10.1006/jnth.1996.0068 | doi-access=free }}
  • {{cite journal | first1=Florian | last1=Luca |title=Perfect Fibonacci and Lucas numbers | year=2000 |journal = Rend. Circ Matem. Palermo |doi=10.1007/BF02904236 | volume=49 | number=2 | pages=313–318 | s2cid=121789033
  • {{cite journal | last = Yabuta | first = M. | journal = Fibonacci Quarterly | pages = 439–443 | title = A simple proof of Carmichael's theorem on primitive divisors | url = http://www.fq.math.ca/Scanned/39-5/yabuta.pdf | volume = 39 | year = 2001 | issue = 5 | doi = 10.1080/00150517.2001.12428701
  • {{cite book | title = Proofs that Really Count: The Art of Combinatorial Proof | first1 = Arthur T. | last1 = Benjamin | author1-link = Arthur T. Benjamin | first2 = Jennifer J. | last2 = Quinn | author2-link = Jennifer Quinn | page = 35 | publisher = Mathematical Association of America | series = Dolciani Mathematical Expositions | volume = 27 | year = 2003 | isbn = 978-0-88385-333-7 | title-link = Proofs That Really Count
  • Lucas sequence at Encyclopedia of Mathematics.

References

  1. "A simpler alternative to Lucas–Lehmer–Riesel primality test".
  2. For such relations and divisibility properties, see {{harv. Carmichael. 1913, {{harv. Lehmer. 1930 or {{harv. Ribenboim. 1996
  3. (2001). "A simple proof of Carmichael's theorem on primitive divisors". Fibonacci Quarterly.
  4. "Primality Proving 3.2 n+1 tests and the Lucas-Lehmer test".
  5. John Brillhart. (April 1975). "New Primality Criteria and Factorizations of 2m ± 1". Mathematics of Computation.
  6. (1993). "LUC: A new public key system". Proceedings of the Ninth IFIP Int. Symp. On Computer Security.
  7. (1995). "Advances in Cryptology — CRYPT0' 95".
  8. "Combinatorial Functions - Combinatorics".

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