Lucas sequence

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:

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

and

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

It is not hard to show that for ${\displaystyle n>0}$,

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

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

${\displaystyle {\begin{array}{r|l|l}n&U_{n}(P,Q)&V_{n}(P,Q)\\\hline 0&0&2\\1&1&P\\2&P&{P}^{2}-2Q\\3&{P}^{2}-Q&{P}^{3}-3PQ\\4&{P}^{3}-2PQ&{P}^{4}-4{P}^{2}Q+2{Q}^{2}\\5&{P}^{4}-3{P}^{2}Q+{Q}^{2}&{P}^{5}-5{P}^{3}Q+5P{Q}^{2}\\6&{P}^{5}-4{P}^{3}Q+3P{Q}^{2}&{P}^{6}-6{P}^{4}Q+9{P}^{2}{Q}^{2}-2{Q}^{3}\end{array}}}$

The characteristic equation of the recurrence relation for Lucas sequences ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ is:

${\displaystyle x^{2}-Px+Q=0\,}$

It has the discriminant ${\displaystyle D=P^{2}-4Q}$ and the roots:

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

Thus:

${\displaystyle a+b=P\,,}$

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

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

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

The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers ${\displaystyle F_{n}=U_{n}(1,-1)}$ and Lucas numbers ${\displaystyle L_{n}=V_{n}(1,-1)}$. For example:

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

Among the consequences is that ${\displaystyle U_{km}(P,Q)}$ is a multiple of ${\displaystyle U_{m}(P,Q)}$, i.e., the sequence ${\displaystyle (U_{m}(P,Q))_{m\geq 1}}$ is a divisibility sequence. This implies, in particular, that ${\displaystyle U_{n}(P,Q)}$ can be prime only when n is prime. Another consequence is an analog of exponentiation by squaring that allows fast computation of ${\displaystyle U_{n}(P,Q)}$ for large values of n. Moreover, if ${\displaystyle \gcd(P,Q)=1}$, then ${\displaystyle (U_{m}(P,Q))_{m\geq 1}}$ is a strong divisibility sequence.

Other divisibility properties are as follows:[1]

• If n / m is odd, then ${\displaystyle V_{m}}$ divides ${\displaystyle V_{n}}$.
• Let N be an integer relatively prime to 2Q. If the smallest positive integer r for which N divides ${\displaystyle U_{r}}$ exists, then the set of n for which N divides ${\displaystyle U_{n}}$ is exactly the set of multiples of r.
• If P and Q are even, then ${\displaystyle U_{n},V_{n}}$ are always even except ${\displaystyle U_{1}}$.
• If P is even and Q is odd, then the parity of ${\displaystyle U_{n}}$ is the same as n and ${\displaystyle V_{n}}$ is always even.
• If P is odd and Q is even, then ${\displaystyle U_{n},V_{n}}$ are always odd for ${\displaystyle n=1,2,\ldots }$.
• If P and Q are odd, then ${\displaystyle U_{n},V_{n}}$ are even if and only if n is a multiple of 3.
• If p is an odd prime, then ${\displaystyle U_{p}\equiv \left({\frac {D}{p}}\right),V_{p}\equiv P{\pmod {p}}}$ (see Legendre symbol).
• If p is an odd prime and divides P and Q, then p divides ${\displaystyle U_{n}}$ for every ${\displaystyle n>1}$.
• If p is an odd prime and divides P but not Q, then p divides ${\displaystyle U_{n}}$ if and only if n is even.
• If p is an odd prime and divides not P but Q, then p never divides ${\displaystyle U_{n}}$ for ${\displaystyle n=1,2,\ldots }$.
• If p is an odd prime and divides not PQ but D, then p divides ${\displaystyle U_{n}}$ if and only if p divides n.
• If p is an odd prime and does not divide PQD, then p divides ${\displaystyle U_{l}}$, where ${\displaystyle l=p-\left({\frac {D}{p}}\right)}$.

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 ${\displaystyle U_{l}}$, where ${\displaystyle l=n-\left({\frac {D}{n}}\right)}$. 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.[2] Indeed, Carmichael (1913) showed that if D is positive and n is not 1, 2 or 6, then ${\displaystyle U_{n}}$ has a primitive prime factor. In the case D is negative, a deep result of Bilu, Hanrot, Voutier and Mignotte[3] shows that if n > 30, then ${\displaystyle U_{n}}$ has a primitive prime factor and determines all cases ${\displaystyle U_{n}}$ has no primitive prime factor.

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.[2]

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 base x

V_n(x+1, x) : xⁿ + 1

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

${\displaystyle P\,}$	${\displaystyle Q\,}$	${\displaystyle U_{n}(P,Q)\,}$	${\displaystyle V_{n}(P,Q)\,}$
−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
2	2	OEIS: A009545	OEIS: A007395
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

• 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[4]
• LUC is a public-key cryptosystem based on Lucas sequences[5] 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.[6] 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 pseudoprime
• Frobenius pseudoprime
• Somer–Lucas pseudoprime

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• 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. hdl:10338.dmlcz/137477. MR 0064073.
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• Joye, M.; Quisquater, J.-J. (1996). "Efficient computation of full Lucas sequences" (PDF). Electronics Letters. 32 (6): 537–538. doi:10.1049/el:19960359. Archived from the original (PDF) on 2015-02-02.
• 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. S2CID 121789033.
• Yabuta, M. (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly. 39: 439–443.
• Benjamin, Arthur T.; Quinn, Jennifer J. (2003). Proofs that Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. 27. Mathematical Association of America. p. 35. ISBN 978-0-88385-333-7.
• Lucas sequence at Encyclopedia of Mathematics.
• Weisstein, Eric W. "Lucas Sequence". MathWorld.
• Wei Dai. "Lucas Sequences in Cryptography".