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 (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:

${\begin{aligned}U_{0}(P,Q)&=0,\\U_{1}(P,Q)&=1,\\U_{n}(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q){\mbox{ for }}n>1,\end{aligned}}$

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 $n>0$,

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

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

${\begin{bmatrix}U_{n}(P,Q)\\U_{n+1}(P,Q)\end{bmatrix}}={\begin{bmatrix}0&1\\-Q&P\end{bmatrix}}\cdot {\begin{bmatrix}U_{n-1}(P,Q)\\U_{n}(P,Q)\end{bmatrix}},$

${\begin{bmatrix}V_{n}(P,Q)\\V_{n+1}(P,Q)\end{bmatrix}}={\begin{bmatrix}0&1\\-Q&P\end{bmatrix}}\cdot {\begin{bmatrix}V_{n-1}(P,Q)\\V_{n}(P,Q)\end{bmatrix}},$

${\begin{bmatrix}U_{n}(P,Q)\\V_{n}(P,Q)\end{bmatrix}}={\begin{bmatrix}P/2&1/2\\(P^{2}-4Q)/2&P/2\end{bmatrix}}\cdot {\begin{bmatrix}U_{n-1}(P,Q)\\V_{n-1}(P,Q)\end{bmatrix}}.$

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

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

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

Pell equations

When $Q=\pm 1$, the Lucas sequences $U_{n}(P,Q)$ and $V_{n}(P,Q)$ satisfy certain Pell equations:

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

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

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

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

$U_{n}(cP,c^{2}Q)=c^{n-1}\cdot U_{n}(P,Q),$

$V_{n}(cP,c^{2}Q)=c^{n}\cdot V_{n}(P,Q).$

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:

${\begin{array}{r|l}{\text{General case}}&(P,Q)=(1,-1)\\\hline (P^{2}-4Q)U_{n}={V_{n+1}-QV_{n-1}}=2V_{n+1}-PV_{n}&5F_{n}={L_{n+1}+L_{n-1}}=2L_{n+1}-L_{n}\\V_{n}=U_{n+1}-QU_{n-1}=2U_{n+1}-PU_{n}&L_{n}=F_{n+1}+F_{n-1}=2F_{n+1}-F_{n}\\U_{2n}=U_{n}V_{n}&F_{2n}=F_{n}L_{n}\\V_{2n}=V_{n}^{2}-2Q^{n}&L_{2n}=L_{n}^{2}-2(-1)^{n}\\U_{m+n}=U_{n}U_{m+1}-QU_{m}U_{n-1}={\frac {U_{m}V_{n}+U_{n}V_{m}}{2}}&F_{m+n}=F_{n}F_{m+1}+F_{m}F_{n-1}={\frac {F_{m}L_{n}+F_{n}L_{m}}{2}}\\V_{m+n}=V_{m}V_{n}-Q^{n}V_{m-n}=DU_{m}U_{n}+Q^{n}V_{m-n}&L_{m+n}=L_{m}L_{n}-(-1)^{n}L_{m-n}=5F_{m}F_{n}+(-1)^{n}L_{m-n}\\V_{n}^{2}-DU_{n}^{2}=4Q^{n}&L_{n}^{2}-5F_{n}^{2}=4(-1)^{n}\\U_{n}^{2}-U_{n-1}U_{n+1}=Q^{n-1}&F_{n}^{2}-F_{n-1}F_{n+1}=(-1)^{n-1}\\V_{n}^{2}-V_{n-1}V_{n+1}=DQ^{n-1}&L_{n}^{2}-L_{n-1}L_{n+1}=5(-1)^{n-1}\\DU_{n}=V_{n+1}-QV_{n-1}&F_{n}={\frac {L_{n+1}+L_{n-1}}{5}}\\V_{m+n}={\frac {V_{m}V_{n}+DU_{m}U_{n}}{2}}&L_{m+n}={\frac {L_{m}L_{n}+5F_{m}F_{n}}{2}}\\U_{m+n}=U_{m}V_{n}-Q^{n}U_{m-n}&F_{n+m}=F_{m}L_{n}-(-1)^{n}F_{m-n}\\2^{n-1}U_{n}={n \choose 1}P^{n-1}+{n \choose 3}P^{n-3}D+\cdots &2^{n-1}F_{n}={n \choose 1}+5{n \choose 3}+\cdots \\2^{n-1}V_{n}=P^{n}+{n \choose 2}P^{n-2}D+{n \choose 4}P^{n-4}D^{2}+\cdots &2^{n-1}L_{n}=1+5{n \choose 2}+5^{2}{n \choose 4}+\cdots \end{array}}$

Divisibility properties

Among the consequences is that $U_{km}(P,Q)$ is a multiple of $U_{m}(P,Q)$, i.e., the sequence $(U_{m}(P,Q))_{m\geq 1}$ 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. Moreover, if $\gcd(P,Q)=1$, then $(U_{m}(P,Q))_{m\geq 1}$ 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 $n>1$.
• 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. 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:

U_n(1, −1) : Fibonacci numbers

V_n(1, −1) : Lucas numbers

U_n(2, −1) : Pell numbers

V_n(2, −1) : Pell–Lucas numbers (companion Pell 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

U_n(6, 1) : The square roots of the square triangular numbers.

U_n(x, −1) : Fibonacci polynomials

V_n(x, −1) : Lucas polynomials

U_n(2x, 1) : Chebyshev polynomials of second kind

V_n(2x, 1) : Chebyshev polynomials of first kind multiplied by 2

U_n(x+1, x) : Repunits in 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	OEIS: A214733
1	−1	OEIS: A000045	OEIS: A000032
1	1	OEIS: A128834	OEIS: A087204
1	2	OEIS: A107920	OEIS: A002249
2	−1	OEIS: A000129	OEIS: A002203
2	1	OEIS: A001477	OEIS: A007395
2	2	OEIS: A009545
2	3	OEIS: A088137
2	4	OEIS: A088138
2	5	OEIS: A045873
3	−5	OEIS: A015523	OEIS: A072263
3	−4	OEIS: A015521	OEIS: A201455
3	−3	OEIS: A030195	OEIS: A172012
3	−2	OEIS: A007482	OEIS: A206776
3	−1	OEIS: A006190	OEIS: A006497
3	1	OEIS: A001906	OEIS: A005248
3	2	OEIS: A000225	OEIS: A000051
3	5	OEIS: A190959
4	−3	OEIS: A015530	OEIS: A080042
4	−2	OEIS: A090017
4	−1	OEIS: A001076	OEIS: A014448
4	1	OEIS: A001353	OEIS: A003500
4	2	OEIS: A007070	OEIS: A056236
4	3	OEIS: A003462	OEIS: A034472
4	4	OEIS: A001787
5	−3	OEIS: A015536
5	−2	OEIS: A015535
5	−1	OEIS: A052918	OEIS: A087130
5	1	OEIS: A004254	OEIS: A003501
5	4	OEIS: A002450	OEIS: A052539
6	1	OEIS: A001109	OEIS: A003499

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.
• 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, a paper by Bleichenbacher et al. 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.

Software

Sagemath implements $U_{n}$ and $V_{n}$ as lucas_number1() and lucas_number2(), respectively.

See also

• Lucas pseudoprime
• Frobenius pseudoprime
• Somer–Lucas pseudoprime

References

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