Lucas sequence

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Not to be confused with the sequence of Lucas numbers, which is a particular Lucas sequence.

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

x_n = P x_n−1 − Q x_n−2

where P and Q are fixed integers. Any other 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. 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:

$U_{0}(P,Q)=0,\,$

$U_{1}(P,Q)=1,\,$

$U_{n}(P,Q)=P\cdot U_{{n-1}}(P,Q)-Q\cdot U_{{n-2}}(P,Q){\mbox{ for }}n>1,\,$

and

$V_{0}(P,Q)=2,\,$

$V_{1}(P,Q)=P,\,$

$V_{n}(P,Q)=P\cdot V_{{n-1}}(P,Q)-Q\cdot V_{{n-2}}(P,Q){\mbox{ for }}n>1,\,$

It is not hard to show that for $n>0$ ,

$U_{n}(P,Q)={\frac {P\cdot U_{{n-1}}(P,Q)+V_{{n-1}}(P,Q)}{2}},\,$

$V_{n}(P,Q)={\frac {(P^{2}-4Q)\cdot U_{{n-1}}(P,Q)+P\cdot V_{{n-1}}(P,Q)}{2}}.\,$

Examples

Initial terms of Lucas sequences U_n(P,Q) and V_n(P,Q) are given in the table:

$n\,$	$U_{n}(P,Q)\,$	$V_{n}(P,Q)\,$
$0\,$	$0\,$	$2\,$
$1\,$	$1\,$	$P\,$
$2\,$	$P\,$	${P}^{{2}}-2Q\,$
$3\,$	${P}^{{2}}-Q\,$	${P}^{{3}}-3PQ\,$
$4\,$	${P}^{{3}}-2PQ\,$	${P}^{{4}}-4{P}^{{2}}Q+2{Q}^{{2}}\,$
$5\,$	${P}^{{4}}-3{P}^{{2}}Q+{Q}^{{2}}\,$	${P}^{{5}}-5{P}^{{3}}Q+5P{Q}^{{2}}\,$
$6\,$	${P}^{{5}}-4{P}^{{3}}Q+3P{Q}^{{2}}\,$	${P}^{{6}}-6{P}^{{4}}Q+9{P}^{{2}}{Q}^{{2}}-2{Q}^{{3}}\,$

Algebraic relations

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 the roots:

$a={\frac {P+{\sqrt {D}}}2}\quad {\text{and}}\quad b={\frac {P-{\sqrt {D}}}2}.\,$

Thus:

$a+b=P\,,$

$ab={\frac {1}{4}}(P^{2}-D)=Q\,,$

$a-b={\sqrt {D}}\,.$

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 $D\neq 0$ , 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

$U_{n}(P,Q)=U_{n}(2S,S^{2})=nS^{{n-1}}\,$

$V_{n}(P,Q)=V_{n}(2S,S^{2})=2S^{n}\,$ .

Additional sequences having the same discriminant

If the Lucas sequences $U_{n}(P,Q)$ and $V_{n}(P,Q)$ have discriminant $D=P^{2}-4Q$ , then the sequences based on $P_{2}$ and $Q_{2}$ where

$P_{2}=P+2$

$Q_{2}=P+Q+1$

have the same discriminant: $P_{2}^{2}-4Q_{2}=(P+2)^{2}-4(P+Q+1)=P^{2}-4Q=D$ .

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:

General	P = 1, Q = -1
$(P^{2}-4Q)U_{n}={V_{{n+1}}-QV_{{n-1}}}=2V_{{n+1}}-PV_{n}\,$	$5F_{n}={L_{{n+1}}+L_{{n-1}}}=2L_{{n+1}}-L_{{n}}\,$
$V_{n}=U_{{n+1}}-QU_{{n-1}}=2U_{{n+1}}-PU_{n}\,$	$L_{n}=F_{{n+1}}+F_{{n-1}}=2F_{{n+1}}-F_{n}\,$
$U_{{2n}}=U_{n}V_{n}\,$	$F_{{2n}}=F_{n}L_{n}\,$
$V_{{2n}}=V_{n}^{2}-2Q^{n}\,$	$L_{{2n}}=L_{n}^{2}-2(-1)^{n}\,$
$U_{{n+m}}=U_{n}U_{{m+1}}-QU_{m}U_{{n-1}}={\frac {U_{n}V_{m}+U_{m}V_{n}}{2}}\,$	$F_{{n+m}}=F_{n}F_{{m+1}}+F_{m}F_{{n-1}}={\frac {F_{n}L_{m}+F_{m}L_{n}}{2}}\,$
$V_{{n+m}}=V_{n}V_{m}-Q^{m}V_{{n-m}}\,$	$L_{{n+m}}=L_{n}L_{m}-(-1)^{m}L_{{n-m}}\,$

Among the consequences is that $U_{{km}}(P,Q)$ is a multiple of $U_{m}(P,Q)$ , i.e., the sequence $(U_{m}(P,Q))_{{m\geq 1}}$ is a divisibility sequence. This implies, in particular, that $U_{n}(P,Q)$ can be prime only when n is prime. Another consequence is an analog of exponentiation by squaring that allows fast computation of $U_{n}(P,Q)$ for large values of n. These facts are used in the Lucas–Lehmer primality test.

Carmichael's theorem states that all but finitely many of the terms in a Lucas sequence have a prime factor that does not divide any earlier term in the sequence (Yubuta 2001).

Specific names

The Lucas sequences for some values of P and Q have specific names:

U_n(1,−1) : Fibonacci numbers

V_n(1,−1) : Lucas numbers

U_n(2,−1) : Pell numbers

V_n(2,−1) : Companion Pell numbers or Pell-Lucas numbers

U_n(1,−2) : Jacobsthal numbers

V_n(1,−2) : Jacobsthal-Lucas numbers

U_n(3, 2) : Mersenne numbers 2ⁿ − 1

V_n(3, 2) : Numbers of the form 2ⁿ + 1, which include the Fermat numbers (Yubuta 2001).

U_n(x,−1) : Fibonacci polynomials

V_n(x,−1) : Lucas polynomials.

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	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		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		A056236
4	3	A003462	A034472
4	4	A001787
5	-3	A015536
5	-2	A015535
5	-1		A087130
5	1		A003501
5	4	A002450	A052539

Applications

LUC is a public-key cryptosystem based on Lucas sequences^[1] 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.^[2] 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.

Lucas sequences are used in probabilistic Lucas pseudoprime tests.

References

↑ 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". Lecture Notes in Computer Science 963: 386–396. doi:10.1007/3-540-44750-4_31.

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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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