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. Although $U_{n} (P, Q)$ and $V_{n} (P, Q)$ satisfy the same recurrence relation, they differ on the values of their first two elements and thus differ for subsequent elements as well. 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{matrix} 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) for n > 1, \end{matrix}

and

\begin{matrix} 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) for n > 1. \end{matrix}

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

\begin{matrix} 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} - 4 Q) \cdot U_{n - 1} (P, Q) + P \cdot V_{n - 1} (P, Q)}{2} . \end{matrix}

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

[\begin{matrix} U_{n} (P, Q) \\ U_{n + 1} (P, Q) \end{matrix}] = [\begin{matrix} 0 & 1 \\ - Q & P \end{matrix}] \cdot [\begin{matrix} U_{n - 1} (P, Q) \\ U_{n} (P, Q) \end{matrix}],

[\begin{matrix} V_{n} (P, Q) \\ V_{n + 1} (P, Q) \end{matrix}] = [\begin{matrix} 0 & 1 \\ - Q & P \end{matrix}] \cdot [\begin{matrix} V_{n - 1} (P, Q) \\ V_{n} (P, Q) \end{matrix}],

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

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

\begin{matrix} n & U_{n} (P, Q) & V_{n} (P, Q) \\ 0 & 0 & 2 \\ 1 & 1 & P \\ 2 & P & P^{2} - 2 Q \\ 3 & P^{2} - Q & P^{3} - 3 P Q \\ 4 & P^{3} - 2 P Q & P^{4} - 4 P^{2} Q + 2 Q^{2} \\ 5 & P^{4} - 3 P^{2} Q + Q^{2} & P^{5} - 5 P^{3} Q + 5 P Q^{2} \\ 6 & P^{5} - 4 P^{3} Q + 3 P Q^{2} & P^{6} - 6 P^{4} Q + 9 P^{2} Q^{2} - 2 Q^{3} \end{matrix}

Explicit expressions

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

x^{2} - P x + Q = 0

It has the discriminant $D = P^{2} - 4 Q$ and, by the quadratic formula, has the roots:

a = \frac{P + \sqrt{D}}{2} and b = \frac{P - \sqrt{D}}{2} .

Thus:

a + b = P,

a b = \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 $P = 2 S and Q = S^{2}$ for some integer S so that $a = b = S$ . In this case one easily finds that

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

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

Properties

Generating functions

The ordinary generating functions are

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

\sum_{n \geq 0} V_{n} (P, Q) z^{n} = \frac{2 - P z}{1 - P z + Q z^{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_{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 $U_{n} (P^{'}, Q^{'})$ and $V_{n} (P^{'}, Q^{'})$ with

P^{'} = P + 2 c

Q^{'} = c P + Q + c^{2}

have the same discriminant as

U_{n} (P, Q)

and

V_{n} (P, Q)

:

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

For any number c, we also have

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

V_{n} (c P, 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{matrix} General case & (P, Q) = (1, - 1), D = P^{2} - 4 Q = 5 \\ D U_{n} = V_{n + 1} - Q V_{n - 1} = 2 V_{n + 1} - P V_{n} & 5 F_{n} = L_{n + 1} + L_{n - 1} = 2 L_{n + 1} - L_{n} & (1) \\ V_{n} = U_{n + 1} - Q U_{n - 1} = 2 U_{n + 1} - P U_{n} & L_{n} = F_{n + 1} + F_{n - 1} = 2 F_{n + 1} - F_{n} & (2) \\ U_{m + n} = U_{n} U_{m + 1} - Q U_{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_{2 n} = U_{n} (U_{n + 1} - Q U_{n - 1}) = U_{n} V_{n} & F_{2 n} = F_{n} (F_{n + 1} + F_{n - 1}) = F_{n} L_{n} & (4) \\ U_{2 n + 1} = U_{n + 1}^{2} - Q U_{n}^{2} & F_{2 n + 1} = F_{n + 1}^{2} + F_{n}^{2} & (5) \\ V_{m + n} = V_{m} V_{n} - Q^{n} V_{m - n} = D U_{m} U_{n} + Q^{n} V_{m - n} & L_{m + n} = L_{m} L_{n} - (- 1)^{n} L_{m - n} = 5 F_{m} F_{n} + (- 1)^{n} L_{m - n} & (6) \\ V_{2 n} = V_{n}^{2} - 2 Q^{n} = D U_{n}^{2} + 2 Q^{n} & L_{2 n} = L_{n}^{2} - 2 (- 1)^{n} = 5 F_{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} + D U_{m} U_{n}}{2} & L_{m + n} = \frac{L_{m} L_{n} + 5 F_{m} F_{n}}{2} & (9) \\ V_{n}^{2} - D U_{n}^{2} = 4 Q^{n} & L_{n}^{2} - 5 F_{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} = D Q^{n - 1} & L_{n}^{2} - L_{n - 1} L_{n + 1} = 5 (- 1)^{n - 1} & (12) \\ 2^{n - 1} U_{n} = (\binom{n}{1}) P^{n - 1} + (\binom{n}{3}) P^{n - 3} D + \dots & 2^{n - 1} F_{n} = (\binom{n}{1}) + 5 (\binom{n}{3}) + \dots & (13) \\ 2^{n - 1} V_{n} = P^{n} + (\binom{n}{2}) P^{n - 2} D + (\binom{n}{4}) P^{n - 4} D^{2} + \dots & 2^{n - 1} L_{n} = 1 + 5 (\binom{n}{2}) + 5^{2} (\binom{n}{4}) + \dots & (14) \end{matrix}

Fast computation

Various analogues of exponentiation by squaring can be used to calculate Lucas sequences efficiently, in ⁠ $𝒪 (\log n)$ ⁠ operations. To simultaneously calculate $U_{n} (P, Q)$ and $V_{n} (P, Q)$ , one can use the properties (3) and (4) to perform double-and-add. Alternatively, calculate the matrix exponentiation $M^{n} = (\begin{matrix} 0 & 1 \\ - Q & P \end{matrix})$ to use $(\begin{matrix} U_{n} \\ U_{n + 1} \end{matrix}) = M^{n} (\begin{matrix} 0 \\ 1 \end{matrix})$ and $(\begin{matrix} V_{n} \\ V_{n + 1} \end{matrix}) = M^{n} (\begin{matrix} 2 \\ P \end{matrix}) .$

(6) and (7) allow fast calculation of V independent of U using double-and-add. The relation $V_{m n} = V_{m} (P = V_{n}, Q = Q_{n})$ (which belongs to the section above, "relations between sequences with different parameters") is also useful for this purpose.

Another ⁠ $𝒪 (\log n)$ ⁠ method based on (3) and (6) is the Lucas chain. This method takes fewer operations than double-and-add but requires pre-calculation of chains.

Divisibility properties

Among the consequences is that $U_{k m} (P, Q)$ 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. 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 $U_{p} \equiv (\frac{D}{p}), V_{p} \equiv P (\mod p)$ (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 $l = p - (\frac{D}{p})$ .

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 $l = n - (\frac{D}{n})$ . 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 (2, 1)

: Counting numbers

U n (1, -2)

: Jacobsthal numbers

V n (1, -2)

: Jacobsthal–Lucas numbers

U n (3, 2)

: Mersenne numbers

2 n - 1

V n (3, 2)

: Numbers of the form

2 n + 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 (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 in base

x

V n (x + 1, x)

:

x n + 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 and Lucas–Lehmer–Riesel tests and the 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, it is argued 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 $V_{n} (P, Q) = a^{n} + b^{n}$ , which is a solution to the recurrence $V_{n} (P, Q) = P V_{n - 1} (P, Q) - Q V_{n - 2} (P, Q)$ when ⁠ $a$ ⁠ and ⁠ $b$ ⁠ are the roots of the corresponding quadratic equation ⁠ $z^{2} - P z + Q = 0$ ⁠, generalizes to degree ⁠ $k \geq 1$ ⁠. Specifically, for the recurrence relation $V_{n} (P_{1}, \dots, P_{k}) = \sum_{j = 1}^{k} P_{j} V_{n - j} (P_{1}, \dots, P_{k})$ with integers ⁠ $P_{1}, \dots, P_{k}$ ⁠ and typically with ⁠ $P_{k} \neq 0$ ⁠, let ⁠ $a_{1}, \dots, a_{k}$ ⁠ be the roots of the corresponding polynomial equation $z^{k} - \sum_{j = 1}^{k} P_{j} z^{k - j} = 0.$ Then $V_{n} (P_{1}, \dots, P_{k}) = \sum_{j = 1}^{k} a_{j}^{n}$ is a sequence of integers satisfying the recurrence, as is evidenced by its ordinary generating function, $G_{P_{1}, \dots, P_{k}} (z) = \sum_{n = 0}^{\infty} V_{n} (P_{1}, \dots, P_{k}) z^{n} = \frac{k - \sum_{j = 1}^{k - 1} (k - j) P_{j} z^{j}}{1 - \sum_{j = 1}^{k} P_{j} z^{j}} .$

Software

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

Notes

↑ Atnashev, Pavel. "A simpler alternative to Lucas–Lehmer–Riesel primality test". Cryptology ePrint Archive.
↑ For such relations and divisibility properties, see (Carmichael 1913), (Lehmer 1930) or (Ribenboim 1996, 2.IV).
1 2 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.
↑ 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.
↑ "Primality Proving 3.2 n+1 tests and the Lucas-Lehmer test". t5k.org.
↑ 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.
↑ 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. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help)
↑ 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.
↑ "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. 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.
Weisstein, Eric W. "Lucas Sequence". MathWorld.
Wei Dai. "Lucas Sequences in Cryptography".

[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).

[Yabuta-3] 1 2 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.

[BLS75-6] 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. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help)

[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.