1. Recurrence relations

2. Examples

3. Explicit expressions
     Distinct roots   Repeated root 

4. Properties
    Generating functions  Sequences with the same discriminant  Pell equations 

5. Other relations

6. Specific names

7. Applications

8. See also

9. Notes

10. References

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

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. 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:

and

It is not hard to show that for ,

Examples

Initial terms of Lucas sequences U_n(P,Q) and V_n(P,Q) 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

Sequences with the same discriminant

If the Lucas sequences and have

discriminant , then the sequences based on and where

have the same discriminant: .

Pell equations

When , the Lucas sequences and satisfy certain Pell equations:

Other relations

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

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]

• If n / m is odd, then divides .
• Let N be an integer relatively prime to 2Q. If the smallest positive integer r for which N divides exists, then the set of n for which N divides is exactly the set of multiples of r.
• If P and Q are even, then are always even except .
• If P is even and Q is odd, then the parity of is the same as n and is always even.
• If P is odd and Q is even, then are always odd for .
• If P and Q are odd, then are even if and only if n is a multiple of 3.
• If p is an odd prime, then (see Legendre symbol).
• If p is an odd prime and divides P and Q, then p divides for every .
• If p is an odd prime and divides P but not Q, then p divides if and only if n is even.
• If p is an odd prime and divides not P but Q, then p never divides for .
• If p is an odd prime and divides not PQ but D, then p divides if and only if p divides n.
• If p is an odd prime and does not divide PQD, then p divides , where .

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 , where . 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 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:

{{math|U_n(1,−1)}} : Fibonacci numbers

{{math|V_n(1,−1)}} : Lucas numbers

{{math|U_n(2,−1)}} : Pell numbers

{{math|V_n(2,−1)}} : Pell-Lucas numbers (companion Pell numbers)

{{math|U_n(1,−2)}} : Jacobsthal numbers

{{math|V_n(1,−2)}} : Jacobsthal-Lucas numbers

{{math|U_n(3, 2)}} : Mersenne numbers 2ⁿ − 1

{{math|V_n(3, 2)}} : Numbers of the form 2ⁿ + 1, which include the Fermat numbers {{harv|Yabuta|2001}}.

{{math|U_n(6, 1)}} : The square roots of the square triangular numbers.

{{math|U_n(x,−1)}} : Fibonacci polynomials

{{math|V_n(x,−1)}} : Lucas polynomials

{{math|U_n(2x, 1)}} : Chebyshev polynomials of second kind

{{math|V_n(2x, 1)}} : Chebyshev polynomials of first kind multiplied by 2

{{math|U_n(x+1, x)}} : Repunits base x

{{math|V_n(x+1, x)}} : xⁿ + 1

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

−1	3	A214733}}
1	−1	A000045}}	A000032}}
1	1	A128834}}	A087204}}
1	2	A107920}}	A002249}}
2	−1	A000129}}	A002203}}
2	1	A001477}}
2	2	A009545}}	A007395}}
2	3	A088137}}
2	4	A088138}}
2	5	A045873}}
3	−5	A015523}}	A072263}}
3	−4	A015521}}	A201455}}
3	−3	A030195}}	A172012}}
3	−2	A007482}}	A206776}}
3	−1	A006190}}	A006497}}
3	1	A001906}}	A005248}}
3	2	A000225}}	A000051}}
3	5	A190959}}
4	−3	A015530}}	A080042}}
4	−2	A090017}}
4	−1	A001076}}	A014448}}
4	1	A001353}}	A003500}}
4	2	A007070}}	A056236}}
4	3	A003462}}	A034472}}
4	4	A001787}}
5	−3	A015536}}
5	−2	A015535}}
5	−1	A052918}}	A087130}}
5	1	A004254}}	A003501}}
5	4	A002450}}	A052539}}
6	1	A001109}}	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^[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.

See also

• Somer–Lucas pseudoprime

Notes

References

• {{citation

• {{cite journal| first1=D. H. | last1=Lehmer

• {{cite journal| first1=Morgan | last1=Ward

• {{cite journal|first1=Lawrence | last1=Somer

• {{cite journal|first1=J. C. | last1=Lagarias

• {{cite book | title=Prime Numbers and Computer Methods for Factorization | edition=2nd | author=Hans Riesel | authorlink=Hans Riesel | series=Progress in Mathematics | volume=126 | publisher=Birkhäuser | year=1994 | isbn=0-8176-3743-5 | pages=107–121 |ref=harv}}
• {{ cite journal|first1=Paulo | last1=Ribenboim | first2=Wayne L. |last2=McDaniel

• {{cite journal | first1=M. | last1=Joye | first2=J.-J. | last2=Quisquater | title=Efficient computation of full Lucas sequences | journal=Electronics Letters | year=1996 | volume=32 | number=6 | pages=537–538 | url=http://www.joye.site88.net/papers/JQ96lucas.pdf | doi=10.1049/el:19960359 | deadurl=yes | archiveurl=https://web.archive.org/web/20150202074230/http://www.joye.site88.net/papers/JQ96lucas.pdf | archivedate=2015-02-02 |ref=harv}}
• {{cite book |first= Paulo |last= Ribenboim |title=The New Book of Prime Number Records | publisher=Springer-Verlag, New York | edition=eBook | isbn=978-1-4612-0759-7 | DOI=10.1007/978-1-4612-0759-7 | year=1996|ref=harv}}
• {{cite book | first=Paulo | last=Ribenboim | authorlink=Paulo Ribenboim | year=2000 | title=My Numbers, My Friends: Popular Lectures on Number Theory | edition= | publisher=Springer-Verlag | location=New York | isbn=0-387-98911-0 | pages=1–50 |ref=harv}}
• {{cite journal | first1=Florian | last1=Luca

• {{cite journal

• {{cite book

• [https://www.encyclopediaofmath.org/index.php/Lucas_sequence Lucas sequence] at Encyclopedia of Mathematics.
• {{MathWorld | urlname=LucasSequence | title=Lucas Sequence}}
• {{cite web| url = http://weidai.com/lucas.html|author=Wei Dai|title= Lucas Sequences in Cryptography}}

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