Lucas sequence explained

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 ⋅ x_n-Q ⋅ x_n

where

and

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

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

Given two integer parameters

and

, 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 ⋅}U_n-1(P,Q)-Q ⋅ U_n-2(P,Q)forn>1, \end{align}

and

\begin{align} V_0(P,Q)&=2,\\ V_1(P,Q)&=P,\\ V_{n(P,Q)&=P ⋅}V_n-1(P,Q)-Q ⋅ V_n-2(P,Q)forn>1. \end{align}

It is not hard to show that for

n>0

\begin{align} U

n(P,Q)&=	P ⋅ U_n-1(P,Q)+V_n-1(P,Q)
	2

\\ V

n(P,Q)&=	(P^{2-4Q) ⋅}U_n-1(P,Q)+P ⋅ 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} ⋅ \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} ⋅ \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} ⋅ \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}²Q+2{Q}²\\ 5&{P}⁴-3{P}²Q+{Q}²&{P}⁵-5{P}³Q+5P{Q}²\\ 6&{P}⁵-4{P}³Q+3P{Q}²&{P}⁶-6{P}⁴Q+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:

x²-Px+Q=0

It has the discriminant

D=P²-4Q

and, by the quadratic formula, has the roots:

	P+\sqrt{D

}2\quad\text\quad b = \frac2. \,

Thus:

a+b=P,

ab=

	1
	4

(P²-D)=Q,

a-b=\sqrt{D}.

Note that the sequence

aⁿ

and the sequence

bⁿ

also satisfy the recurrence relation. However these might not be integer sequences.

Distinct roots

When

D\ne0

, a and b are distinct and one quickly verifies that

aⁿ=

	V_n+U_n\sqrt{D

}

bⁿ=

	V_n-U_n\sqrt{D

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

U_n=

	a^n-bⁿ
	a-b

	a^n-bⁿ
	\sqrt{D

}

V_n=a^n+bⁿ

Repeated root

The case

D=0

occurs exactly when

P=2SandQ=S²

for some integer S so that

a=b=S

. In this case one easily finds that

U_n(P,Q)=U

	2)

	n(2S,S

=nS^n-1

V_n(P,Q)=V

	2)=2S

	n(2S,S

^n.

Properties

Generating functions

The ordinary generating functions are

\sum_n\ge

	n
U
	n(P,Q)z

	z
	1-Pz+Qz²

;

\sum_n\ge

	n
V
	n(P,Q)z

	2-Pz
	1-Pz+Qz²

Pell equations

When

Q=\pm1

, the Lucas sequences

U_n(P,Q)

and

V_n(P,Q)

satisfy certain Pell equations:

	2
V
	n(P,1)

-D ⋅

	2
U
	n(P,1)

=4,

	2
V
	n(P,-1)

-D ⋅

	2
U
	n(P,-1)

=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+c²

have the same discriminant as

U_n(P,Q)

and

V_n(P,Q)

P'²-4Q'=(P+2c)²-4(cP+Q+c²⁾=P²-4Q=D.

For any number c, we also have

	2Q)
U
	n(cP,c

=c^n-1 ⋅ U_n(P,Q),

	2Q)
V
	n(cP,c

=c^{n ⋅}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} Generalcase&(P,Q)=(1,-1),D=P²-4Q=5 \\ \hline DU_n={V_n+1-QV_n-1

}=2V_-P V_n & 5F_n = =2L_ - L_ & (1)\\V_n = U_ - Q U_=2U_-PU_n & L_n = F_ + F_=2F_-F_n & (2)\\U_ = U_n U_ - Q U_m U_ = U_mV_n-Q^nU_ & F_ = F_n F_ + F_m F_ =F_mL_n-(-1)^nF_ & (3)\\U_ = U_n (U_ - QU_) = U_n V_n & F_ = F_n (F_ + F_) = F_n L_n & (4)\\U_ = U_^2 - Q U_n^2 & F_ = F_^2 + F_n^2 & (5)\\V_ = V_m V_n - Q^n V_ = D U_m U_n + Q^n V_ & L_ = L_m L_n - (-1)^n L_ = 5 F_m F_n + (-1)^n L_ & (6)\\V_ = V_n^2 - 2Q^n = D U_n^2 + 2Q^n & L_ = L_n^2 - 2(-1)^n = 5 F_n^2 + 2(-1)^n & (7)\\U_ = \frac & F_ = \frac & (8)\\V_=\frac & L_=\frac & (9)\\V_n^2-DU_n^2=4Q^n & L_n^2-5F_n^2=4(-1)^n & (10)\\U_n^2-U_U_=Q^ & F_n^2-F_F_=(-1)^ & (11)\\V_n^2-V_V_=DQ^ & L_n^2-L_L_=5(-1)^ & (12)\\2^U_n=P^+P^D+\cdots & 2^F_n=+5+\cdots & (13)\\2^V_n=P^n+P^D+P^D^2+\cdots & 2^L_n=1+5+5^2+\cdots & (14)\end

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.^[1]

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:^[2]

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

U₁

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\equivP\pmod{p}

(see Legendre symbol).

If p is an odd prime which divides P and Q, then p divides

U_n

for every

n>1

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.^[3] 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^[4] 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 and have specific names:

: Fibonacci numbers

: Lucas numbers

: Pell numbers

: Pell–Lucas numbers (companion Pell numbers)

: Counting numbers

: Jacobsthal numbers

: Jacobsthal–Lucas numbers

: Mersenne numbers

: Numbers of the form, which include the Fermat numbers^[3]

: The square roots of the square triangular numbers.

: Fibonacci polynomials

: Lucas polynomials

: Chebyshev polynomials of second kind

: Chebyshev polynomials of first kind multiplied by 2

: Repunits in base

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

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.^[5] ^[6]
LUC is a public-key cryptosystem based on Lucas sequences^[7] 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^[8] 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)=aⁿ+bⁿ

, which is a solution to the recurrence

V_n(P,Q)=PV_n-1(P,Q)-QV_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_^k P_j V_(P_1, \ldots, P_k)

with integers and typically with, let be the roots of the corresponding polynomial equation

z^k - \sum_^k P_j z^ = 0.

Then

V_n(P_1, \ldots, P_k) = \sum_^k a_j^n

is a sequence of integers satisfying the recurrence, as is evidenced by its ordinary generating function,

G_(z) = \sum_^\infty V_n(P_1, \ldots, P_k) z^n = \frac.

Software

SageMath implements

U_n

and

V_n

as functions lucas_number1 and lucas_number2, respectively.^[9]

References

- D. H. . Lehmer. An extended theory of Lucas' functions. Annals of Mathematics . 1930. 31 . 3. 1968235 . 419–448 . 1930AnMat..31..419L . 10.2307/1968235.
Morgan . Ward. Prime divisors of second order recurring sequences. Duke Math. J. . 1954 . 21 . 4. 607–614 . 0064073 . 10.1215/S0012-7094-54-02163-8. 10338.dmlcz/137477. free.
Lawrence . Somer. The divisibility properties of primary Lucas Recurrences with respect to primes. 1980 . Fibonacci Quarterly . 316–334 . 18 . 4. 10.1080/00150517.1980.12430140.
J. C. . Lagarias. Pac. J. Math. . The set of primes dividing Lucas Numbers has density 2/3. 1985 . 118 . 2 . 449–461 . 789184 . 10.2140/pjm.1985.118.449. 10.1.1.174.660 .
Book: Prime Numbers and Computer Methods for Factorization . 2nd . Hans Riesel . Hans Riesel . Progress in Mathematics . 126 . Birkhäuser . 1994 . 0-8176-3743-5 . 107–121 .
Paulo . Ribenboim . Wayne L. . McDaniel. The square terms in Lucas Sequences . J. Number Theory . 1996 . 58 . 1 . 104–123 . 10.1006/jnth.1996.0068. free .
M. . Joye . J.-J. . Quisquater . Efficient computation of full Lucas sequences . Electronics Letters . 1996 . 32 . 6 . 537–538 . 10.1049/el:19960359 . 1996ElL....32..537J . dead . https://web.archive.org/web/20150202074230/http://www.joye.site88.net/papers/JQ96lucas.pdf . 2015-02-02 .
Book: Ribenboim, Paulo . The New Book of Prime Number Records . Springer-Verlag, New York . eBook . 978-1-4612-0759-7 . 10.1007/978-1-4612-0759-7 . 1996.
Book: Ribenboim, Paulo . Paulo Ribenboim . 2000 . My Numbers, My Friends: Popular Lectures on Number Theory . . New York . 0-387-98911-0 . 1–50 .
Florian . Luca. Perfect Fibonacci and Lucas numbers . 2000. Rend. Circ Matem. Palermo . 10.1007/BF02904236 . 49 . 2 . 313–318. 121789033.
Yabuta . M. . Fibonacci Quarterly . 439–443 . A simple proof of Carmichael's theorem on primitive divisors . 39 . 2001 . 5 . 10.1080/00150517.2001.12428701 .
Book: Proofs that Really Count: The Art of Combinatorial Proof . Arthur T. . Benjamin . Arthur T. Benjamin . Jennifer J. . Quinn . Jennifer Quinn . 35 . . Dolciani Mathematical Expositions . 27 . 2003 . 978-0-88385-333-7 . Proofs That Really Count .
Lucas sequence at Encyclopedia of Mathematics.
- Web site: Lucas Sequences in Cryptography. Wei Dai. Wei Dai.

Notes and References

Web site: Atnashev . Pavel . A simpler alternative to Lucas–Lehmer–Riesel primality test . Cryptology ePrint Archive.
For such relations and divisibility properties, see, or .
Yabuta . M . A simple proof of Carmichael's theorem on primitive divisors . Fibonacci Quarterly . 2001 . 39 . 5 . 439–443 . 10.1080/00150517.2001.12428701 . 4 October 2018.
Yuri . Bilu . Guillaume . Hanrot . Paul M. . Voutier . Maurice . Mignotte . Existence of primitive divisors of Lucas and Lehmer numbers . J. Reine Angew. Math. . 2001 . 2001 . 539 . 75–122 . 1863855 . 10.1515/crll.2001.080 . 122969549 .
Web site: Primality Proving 3.2 n+1 tests and the Lucas-Lehmer test . t5k.org.
John Brillhart. Derrick Henry Lehmer. John Selfridge. John Selfridge. New Primality Criteria and Factorizations of 2^m ± 1. Mathematics of Computation . 29. 130. April 1975. 620–647. 2005583. 10.1090/S0025-5718-1975-0384673-1. John Brillhart. Derrick Henry Lehmer. free.
P. J. Smith . M. J. J. Lennon . LUC: A new public key system . Proceedings of the Ninth IFIP Int. Symp. On Computer Security . 1993 . 103–117 . 10.1.1.32.1835 .
Book: D. Bleichenbacher . W. Bosma . A. K. Lenstra . Advances in Cryptology — CRYPT0' 95 . Some Remarks on Lucas-Based Cryptosystems . Lecture Notes in Computer Science . 963 . 1995 . 386–396 . 10.1007/3-540-44750-4_31 . 978-3-540-60221-7 . http://www.math.ru.nl/~bosma/pubs/CRYPTO95.pdf. free .
Web site: Combinatorial Functions - Combinatorics . 2023-07-13 . doc.sagemath.org.

Lucas sequence explained

Recurrence relations

Explicit expressions

Distinct roots

Repeated root

Properties

Generating functions

Pell equations

Relations between sequences with different parameters

Other relations

Fast computation

Divisibility properties

Specific names

Applications

Generalizations

Software

See also

References

Notes and References