Lucas sequence

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

In mathematics, the Lucas sequences${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ are certain constant-recursive integer sequences that satisfy the recurrence relation

${\displaystyle x_{n}=P\cdot x_{n-1}-Q\cdot x_{n-2}}$

where ${\displaystyle P}$ and ${\displaystyle Q}$ are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q).}$

More generally, Lucas sequences ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ represent sequences of polynomials in ${\displaystyle P}$ and ${\displaystyle Q}$ 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 ${\displaystyle P}$ and ${\displaystyle Q}$, the Lucas sequences of the first kind ${\displaystyle U_{n}(P,Q)}$ and of the second kind ${\displaystyle V_{n}(P,Q)}$ 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 ${\displaystyle n>0}$,

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

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

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

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

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

Initial terms of Lucas sequences ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ are given in the table:

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

Explicit expressions

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, by the quadratic formula, has 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.

Distinct roots

When ${\displaystyle D\neq 0}$, a and b are distinct and one quickly verifies that

${\displaystyle a^{n}={\frac {V_{n}+U_{n}{\sqrt {D}}}{2}}}$

${\displaystyle b^{n}={\frac {V_{n}-U_{n}{\sqrt {D}}}{2}}.}$

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

${\displaystyle U_{n}={\frac {a^{n}-b^{n}}{a-b}}={\frac {a^{n}-b^{n}}{\sqrt {D}}}}$

${\displaystyle V_{n}=a^{n}+b^{n}\,}$

Repeated root

The case ${\displaystyle D=0}$ occurs exactly when ${\displaystyle P=2S{\text{ and }}Q=S^{2}}$ for some integer S so that ${\displaystyle a=b=S}$. In this case one easily finds that

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

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

Properties

Generating functions

The ordinary generating functions are

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

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

Pell equations

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

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

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

Relations between sequences with different parameters

• For any number c, the sequences ${\displaystyle U_{n}(P',Q')}$ and ${\displaystyle V_{n}(P',Q')}$ with

${\displaystyle P'=P+2c}$

${\displaystyle Q'=cP+Q+c^{2}}$

have the same discriminant as ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$:

${\displaystyle P'^{2}-4Q'=(P+2c)^{2}-4(cP+Q+c^{2})=P^{2}-4Q=D.}$

• For any number c, we also have

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

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

Other relations

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

${\displaystyle {\begin{array}{l|l|r}{\text{General case}}&(P,Q)=(1,-1),D=P^{2}-4Q=5\\\hline DU_{n}={V_{n+1}-QV_{n-1}}=2V_{n+1}-PV_{n}&5F_{n}={L_{n+1}+L_{n-1}}=2L_{n+1}-L_{n}&(1)\\V_{n}=U_{n+1}-QU_{n-1}=2U_{n+1}-PU_{n}&L_{n}=F_{n+1}+F_{n-1}=2F_{n+1}-F_{n}&(2)\\U_{m+n}=U_{n}U_{m+1}-QU_{m}U_{n-1}=U_{m}V_{n}-Q^{n}U_{m-n}&F_{m+n}=F_{n}F_{m+1}+F_{m}F_{n-1}=F_{m}L_{n}-(-1)^{n}F_{m-n}&(3)\\U_{2n}=U_{n}(U_{n+1}-QU_{n-1})=U_{n}V_{n}&F_{2n}=F_{n}(F_{n+1}+F_{n-1})=F_{n}L_{n}&(4)\\U_{2n+1}=U_{n+1}^{2}-QU_{n}^{2}&F_{2n+1}=F_{n+1}^{2}+F_{n}^{2}&(5)\\V_{m+n}=V_{m}V_{n}-Q^{n}V_{m-n}=DU_{m}U_{n}+Q^{n}V_{m-n}&L_{m+n}=L_{m}L_{n}-(-1)^{n}L_{m-n}=5F_{m}F_{n}+(-1)^{n}L_{m-n}&(6)\\V_{2n}=V_{n}^{2}-2Q^{n}=DU_{n}^{2}+2Q^{n}&L_{2n}=L_{n}^{2}-2(-1)^{n}=5F_{n}^{2}+2(-1)^{n}&(7)\\U_{m+n}={\frac {U_{m}V_{n}+U_{n}V_{m}}{2}}&F_{m+n}={\frac {F_{m}L_{n}+F_{n}L_{m}}{2}}&(8)\\V_{m+n}={\frac {V_{m}V_{n}+DU_{m}U_{n}}{2}}&L_{m+n}={\frac {L_{m}L_{n}+5F_{m}F_{n}}{2}}&(9)\\V_{n}^{2}-DU_{n}^{2}=4Q^{n}&L_{n}^{2}-5F_{n}^{2}=4(-1)^{n}&(10)\\U_{n}^{2}-U_{n-1}U_{n+1}=Q^{n-1}&F_{n}^{2}-F_{n-1}F_{n+1}=(-1)^{n-1}&(11)\\V_{n}^{2}-V_{n-1}V_{n+1}=DQ^{n-1}&L_{n}^{2}-L_{n-1}L_{n+1}=5(-1)^{n-1}&(12)\\2^{n-1}U_{n}={n \choose 1}P^{n-1}+{n \choose 3}P^{n-3}D+\cdots &2^{n-1}F_{n}={n \choose 1}+5{n \choose 3}+\cdots &(13)\\2^{n-1}V_{n}=P^{n}+{n \choose 2}P^{n-2}D+{n \choose 4}P^{n-4}D^{2}+\cdots &2^{n-1}L_{n}=1+5{n \choose 2}+5^{2}{n \choose 4}+\cdots &(14)\end{array}}}$

Of these, (6) and (7) allow fast calculation of V independent of U in a way analogous to exponentiation by squaring. The relation ${\displaystyle V_{mn}=V_{m}(P=V_{n},Q=Q_{n})}$ (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 ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ from ${\displaystyle U_{n-1}(P,Q)}$ and ${\displaystyle V_{n-1}(P,Q)}$ allows ⁠${\displaystyle {\mathcal {O}}(\log n)}$⁠-time computation of ${\displaystyle U_{n}(P,Q)}$ and ${\displaystyle V_{n}(P,Q)}$ for large values of n.

Divisibility properties

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. 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: ^[2]

• If n is an odd multiple of m, 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 odd and Q is even, then ${\displaystyle U_{n},V_{n}}$ are always odd for every ${\displaystyle n>0}$.
• 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 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 ${\displaystyle U_{p}\equiv \left({\tfrac {D}{p}}\right),V_{p}\equiv P{\pmod {p}}}$ (see Legendre symbol).
• If p is an odd prime which divides P and Q, then p divides ${\displaystyle U_{n}}$ for every ${\displaystyle n>1}$.
• If p is an odd prime which divides P but not Q, then p divides ${\displaystyle U_{n}}$ if and only if n is even.
• If p is an odd prime which divides Q but not P, then p never divides ${\displaystyle U_{n}}$ for any ${\displaystyle n>0}$.
• If p is an odd prime which divides D but not PQ, then p divides ${\displaystyle U_{n}}$ if and only if p divides n.
• If p is an odd prime which does not divide PQD, then p divides ${\displaystyle U_{l}}$, where ${\displaystyle l=p-\left({\tfrac {D}{p}}\right)}$.

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 ${\displaystyle U_{l}}$, where ${\displaystyle l=n-\left({\tfrac {D}{n}}\right)}$. 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 ${\displaystyle U_{n}}$ 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 ${\displaystyle U_{n}}$ has a primitive prime factor and determines all cases ${\displaystyle U_{n}}$ 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(2, 1) : Counting 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 ^[3]

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:

${\displaystyle P\,}$	${\displaystyle Q\,}$	${\displaystyle U_{n}(P,Q)\,}$	${\displaystyle V_{n}(P,Q)\,}$
−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	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 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 ${\displaystyle V_{n}(P,Q)=a^{n}+b^{n}}$, which is a solution to the recurrence ${\displaystyle V_{n}(P,Q)=PV_{n-1}(P,Q)-QV_{n-2}(P,Q)}$ when ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are the roots of the corresponding quadratic equation ⁠${\displaystyle z^{2}-Pz+Q=0}$⁠, generalizes to degree ⁠${\displaystyle k\geq 1}$⁠. Specifically, for the recurrence relation $${\displaystyle V_{n}(P_{1},\ldots ,P_{k})=\sum _{j=1}^{k}P_{j}V_{n-j}(P_{1},\ldots ,P_{k})}$$ with integers ⁠${\displaystyle P_{1},\ldots ,P_{k}}$⁠ and typically with ⁠${\displaystyle P_{k}\neq 0}$⁠, let ⁠${\displaystyle a_{1},\ldots ,a_{k}}$⁠ be the roots of the corresponding polynomial equation $${\displaystyle z^{k}-\sum _{j=1}^{k}P_{j}z^{k-j}=0.}$$ Then $${\displaystyle V_{n}(P_{1},\ldots ,P_{k})=\sum _{j=1}^{k}a_{j}^{n}}$$ is a sequence of integers satisfying the recurrence, as is evidenced by its ordinary generating function, $${\displaystyle G_{P_{1},\ldots ,P_{k}}(z)=\sum _{n=0}^{\infty }V_{n}(P_{1},\ldots ,P_{k})z^{n}={\frac {k-\sum _{j=1}^{k-1}(k-j)P_{j}z^{j}}{1-\sum _{j=1}^{k}P_{j}z^{j}}}.}$$

Software

• SageMath implements ${\displaystyle U_{n}}$ and ${\displaystyle V_{n}}$ as functions lucas_number1() and lucas_number2(), respectively. ^[9]

See also

• Lucas pseudoprime
• Frobenius pseudoprime
• Somer–Lucas pseudoprime

Notes

1. Atnashev, Pavel. "A simpler alternative to Lucas–Lehmer–Riesel primality test". Cryptology ePrint Archive.
2. For such relations and divisibility properties, see ( Carmichael 1913 ), ( Lehmer 1930 ) or ( Ribenboim 1996 , 2.IV).
3. Yabuta, M (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly. 39 (5): 439–443. doi:10.1080/00150517.2001.12428701 . Retrieved 4 October 2018.
4. Bilu, Yuri; Hanrot, Guillaume; Voutier, Paul M.; Mignotte, Maurice (2001). "Existence of primitive divisors of Lucas and Lehmer numbers" (PDF). J. Reine Angew. Math. 2001 (539): 75–122. doi:10.1515/crll.2001.080. MR   1863855. S2CID   122969549.
5. "Primality Proving 3.2 n+1 tests and the Lucas-Lehmer test". t5k.org.
6. 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 . JSTOR   2005583.
7. 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. CiteSeerX   10.1.1.32.1835 .
8. D. Bleichenbacher; W. Bosma; A. K. Lenstra (1995). "Some Remarks on Lucas-Based Cryptosystems" (PDF). Advances in Cryptology — CRYPT0' 95. Lecture Notes in Computer Science. Vol. 963. pp. 386–396. doi: 10.1007/3-540-44750-4_31 . ISBN   978-3-540-60221-7.
9. "Combinatorial Functions - Combinatorics". doc.sagemath.org. Retrieved 2023-07-13.

References

• Carmichael, R. D. (1913), "On the numerical factors of the arithmetic forms αⁿ±βⁿ", Annals of Mathematics, 15 (1/4): 30–70, doi:10.2307/1967797, JSTOR   1967797
• 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. hdl: 10338.dmlcz/137477 . MR   0064073.
• Somer, Lawrence (1980). "The divisibility properties of primary Lucas Recurrences with respect to primes" (PDF). Fibonacci Quarterly. 18 (4): 316–334. doi:10.1080/00150517.1980.12430140.
• Lagarias, J. C. (1985). "The set of primes dividing Lucas Numbers has density 2/3". Pac. J. Math. 118 (2): 449–461. CiteSeerX   10.1.1.174.660 . doi:10.2140/pjm.1985.118.449. MR   0789184.
• Hans Riesel (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Vol. 126 (2nd ed.). Birkhäuser. pp. 107–121. ISBN   0-8176-3743-5.
• Ribenboim, Paulo; McDaniel, Wayne L. (1996). "The square terms in Lucas Sequences". J. Number Theory. 58 (1): 104–123. doi: 10.1006/jnth.1996.0068 .
• Joye, M.; Quisquater, J.-J. (1996). "Efficient computation of full Lucas sequences" (PDF). Electronics Letters. 32 (6): 537–538. Bibcode:1996ElL....32..537J. 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 (5): 439–443. doi:10.1080/00150517.2001.12428701.
• Benjamin, Arthur T.; Quinn, Jennifer J. (2003). Proofs that Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. Vol. 27. Mathematical Association of America. p.  35. ISBN   978-0-88385-333-7.
• Lucas sequence at Encyclopedia of Mathematics.
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