Lucas sequence

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In mathematics, the Lucas sequences and are certain constant-recursive integer sequences that satisfy the recurrence relation

Contents

where and are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences and

More generally, Lucas sequences and represent sequences of polynomials in and 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 and , the Lucas sequences of the first kind and of the second kind are defined by the recurrence relations:

and

It is not hard to show that for ,

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



Examples

Initial terms of Lucas sequences and are given in the table:

Explicit expressions

The characteristic equation of the recurrence relation for Lucas sequences and is:

It has the discriminant and the roots:

Thus:

Note that the sequence and the sequence also satisfy the recurrence relation. However these might not be integer sequences.

Distinct roots

When , a and b are distinct and one quickly verifies that

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

Repeated root

The case occurs exactly when for some integer S so that . In this case one easily finds that

Properties

Generating functions

The ordinary generating functions are

Pell equations

When , the Lucas sequences and satisfy certain Pell equations:

Relations between sequences with different parameters

have the same discriminant as and :

Other relations

The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers and Lucas numbers . For example:

Divisibility properties

Among the consequences is that is a multiple of , i.e., the sequence is a divisibility sequence. This implies, in particular, that can be prime only when n is prime. Another consequence is an analog of exponentiation by squaring that allows fast computation of for large values of n. Moreover, if , then is a strong divisibility sequence.

Other divisibility properties are as follows: [1]

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 , where . 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. [2] Indeed, Carmichael (1913) showed that if D is positive and n is not 1, 2 or 6, then 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 has a primitive prime factor and determines all cases has no primitive prime factor.

Specific names

The Lucas sequences for some values of P and Q have specific names:

Un(1, −1) : Fibonacci numbers
Vn(1, −1) : Lucas numbers
Un(2, −1) : Pell numbers
Vn(2, −1) : Pell–Lucas numbers (companion Pell numbers)
Un(1, −2) : Jacobsthal numbers
Vn(1, −2) : Jacobsthal–Lucas numbers
Un(3, 2) : Mersenne numbers 2n 1
Vn(3, 2) : Numbers of the form 2n +1, which include the Fermat numbers [2]
Un(6,1) : The square roots of the square triangular numbers.
Un(x, −1) : Fibonacci polynomials
Vn(x, −1) : Lucas polynomials
Un(2x,1) : Chebyshev polynomials of second kind
Vn(2x,1) : Chebyshev polynomials of first kind multiplied by 2
Un(x+1, x) : Repunits in base x
Vn(x+1, x) : xn +1

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

−13 OEIS:  A214733
1−1 OEIS:  A000045 OEIS:  A000032
11 OEIS:  A128834 OEIS:  A087204
12 OEIS:  A107920 OEIS:  A002249
2−1 OEIS:  A000129 OEIS:  A002203
21 OEIS:  A001477 OEIS:  A007395
22 OEIS:  A009545
23 OEIS:  A088137
24 OEIS:  A088138
25 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
31 OEIS:  A001906 OEIS:  A005248
32 OEIS:  A000225 OEIS:  A000051
35 OEIS:  A190959
4−3 OEIS:  A015530 OEIS:  A080042
4−2 OEIS:  A090017
4−1 OEIS:  A001076 OEIS:  A014448
41 OEIS:  A001353 OEIS:  A003500
42 OEIS:  A007070 OEIS:  A056236
43 OEIS:  A003462 OEIS:  A034472
44 OEIS:  A001787
5−3 OEIS:  A015536
5−2 OEIS:  A015535
5−1 OEIS:  A052918 OEIS:  A087130
51 OEIS:  A004254 OEIS:  A003501
54 OEIS:  A002450 OEIS:  A052539
61 OEIS:  A001109 OEIS:  A003499

Applications

Software

See also

Notes

  1. For such relations and divisibility properties, see ( Carmichael 1913 ), ( Lehmer 1930 ) or ( Ribenboim 1996 , 2.IV).
  2. 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.
  3. 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.
  4. John Brillhart; Derrick Henry Lehmer; John Selfridge (April 1975). "New Primality Criteria and Factorizations of 2m ± 1". Mathematics of Computation. 29 (130): 620–647. doi: 10.1090/S0025-5718-1975-0384673-1 . JSTOR   2005583.
  5. 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 .
  6. 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.
  7. "Combinatorial Functions - Combinatorics". doc.sagemath.org. Retrieved 2023-07-13.

References