Lucas sequence

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 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. 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{aligned}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,\end{aligned}}

and

{\begin{aligned}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.\end{aligned}}

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

{\begin{aligned}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{aligned}}

Examples

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 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}\,

Properties

Generating functions

The ordinary generating functions are

\sum _{n\geq 0}U_{n}(P,Q)z^{n}={\frac {z}{1-Pz+Qz^{2}}};

\sum _{n\geq 0}V_{n}(P,Q)z^{n}={\frac {2-Pz}{1-Pz+Qz^{2}}}.

Sequences with 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: $P_{2}^{2}-4Q_{2}=(P+2)^{2}-4(P+Q+1)=P^{2}-4Q=D$ .

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_{2n}(P,-1)^{2}-D\cdot U_{2n}(P,-1)^{2}=4,

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

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}{r|l}{\text{General case}}&(P,Q)=(1,-1)\\\hline (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_{m+n}=U_{n}U_{m+1}-QU_{m}U_{n-1}={\frac {U_{m}V_{n}+U_{n}V_{m}}{2}}&F_{m+n}=F_{n}F_{m+1}+F_{m}F_{n-1}={\frac {F_{m}L_{n}+F_{n}L_{m}}{2}}\\V_{m+n}=V_{m}V_{n}-Q^{n}V_{m-n}=DU_{m}U_{n}+Q^{n}V_{m-n}&L_{m+n}=L_{m}L_{n}-(-1)^{n}L_{m-n}=5F_{m}F_{n}+(-1)^{n}L_{m-n}\\V_{n}^{2}-DU_{n}^{2}=4Q^{n}&L_{n}^{2}-5F_{n}^{2}=4(-1)^{n}\\U_{n}^{2}-U_{n-1}U_{n+1}=Q^{n-1}&F_{n}^{2}-F_{n-1}F_{n+1}=(-1)^{n}\\DU_{n}=V_{n+1}-QV_{n-1}&F_{n}={\frac {L_{n+1}+L_{n-1}}{5}}\\V_{m+n}={\frac {V_{m}V_{n}+DU_{m}U_{n}}{2}}&L_{m+n}={\frac {L_{m}L_{n}+5F_{m}F_{n}}{2}}\\U_{m+n}=U_{m}V_{n}-Q^{n}U_{m-n}&F_{n+m}=F_{m}L_{n}-5^{n}F_{m-n}\\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 \\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 \end{array}}

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. Further divisibility properties are follows:^[1]

If n / m is odd, 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 even and Q is odd, then the parity of $U_{n}$ is the same as n and $V_{n}$ is always even.
If P is odd and Q is even, then $U_{n},V_{n}$ are always odd for $n=1,2,\ldots$ .
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 $U_{p}\equiv \left({\frac {D}{p}}\right),V_{p}\equiv P{\pmod {p}}$ .
If p is an odd prime and divides P and Q, then p divides $U_{n}$ for every $n>1$ .
If p is an odd prime and divides P but not Q, then p divides $U_{n}$ if and only if n is even.
If p is an odd prime and divides not P but Q, then p never divides $U_{n}$ for $n=1,2,\ldots$ .
If p is an odd prime and divides not PQ but D, then p divides $U_{n}$ if and only if p divides n.
If p is an odd prime and does not divide PQD, then p divides $U_{l}$ , where $l=p-\left({\frac {D}{p}}\right)$ .

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 $U_{l}$ , where $l=n-\left({\frac {D}{n}}\right)$ . Such a composite is called 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(Yubuta 2001). 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^[2] 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:

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

U n (2 x, 1)

: Chebyshev polynomials of second kind

V n (2 x, 1)

: Chebyshev polynomials of first kind multiplied by 2

U n (x +1, x)

: Repunits base x

V n (x +1, x)

: xⁿ + 1

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	A007482	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	A007070	A056236
4	3	A003462	A034472
4	4	A001787
5	-3	A015536
5	-2	A015535
5	-1	A052918	A087130
5	1	A004254	A003501
5	4	A002450	A052539

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-Riesel test, and the N+1 and hybrid N-1/N+1 methods such as those in Brillhart-Lehmer-Selfridge 1975^[3]
LUC is a public-key cryptosystem based on Lucas sequences^[4] 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.^[5] 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.

Notes

↑ For such relations and divisibility properties, see Carmichael (1913), Lehmer (1930) or 2. IV of Ribenboim (1996).
↑ Bilu, Yuri; Hanrot, Guillaume; Voutier, Paul M.; Mignotte, Maurice (2001). "Existence of primitive divisors of Lucas and Lehmer numbers". J. Reine Angew. Math. 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 2^m ± 1". Mathematics of Computation. 29 (130): 620–647. doi:10.1090/S0025-5718-1975-0384673-1.
↑ 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" (PDF). Lecture Notes in Computer Science. 963: 386–396. doi:10.1007/3-540-44750-4_31.

References

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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. 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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Joye, M.; Quisquater, J.-J. (1996). "Efficient computation of full Lucas sequences" (PDF). El. Lett. 32 (6): 537–538. doi:10.1049/el:19960359.
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.
Luca, Florian (2000). "Perfect Fibonacci and Lucas numbers". Rend. Circ Matem. Palermo. 49 (2): 313–318. doi:10.1007/BF02904236.
Yabuta, M. (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly. 39: 439–443. .
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Wei Dai. "Lucas Sequences in Cryptography".

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