## Generalized Lucas Number |

In a problem in his text *Liber Abbaci* (published in 1202), Fibonacci introduced his now famous sequence:
*U*_{0}, *U*_{1}, *U*_{2},
...} and {*V*_{0}, *V*_{1}, *V*_{2}, ...}
for which the next term is P times the current term minus Q times the previous
one:
*D* =
P^{2}-4Q is also not zero.
*U*_{0}=0, *U*_{1}=1,*V*_{0}=2, and *V*_{1}=P;
and the sequences follow the recurrence relations given above. These sequences
are both called **Lucas sequences**, and the numbers in them are the **generalized
Lucas numbers**.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ....Each term is the sum of the two proceeding terms. Lucas [Lucas1878] generalized this by defining pairs of sequences {

We usually require that P and Q be non-zero integers and that (the discriminant)U_{n+1}= P*U_{n}- Q*U_{n-1}andV_{n+1}= P*V_{n}- Q*V_{n-1}.

To define the Lucas sequences, let *a* and *b* be the zeros of the polynomial *x*^{2}-P*x*+Q, then define the two companion sequences as follows:

SoU_{n}(P,Q) = (a^{n}-b^{n})/(a-b), andV_{n}(P,Q) =a^{n}+b^{n}.

These sequences have many useful properties such as:
*U*_{2n}=*U*_{n}*V*_{n}; and if *p* is and odd prime, then *p* divides *U*_{p-(D/p)} where (*D*/*p*) is the Legendre symbol. Ribenboim's book (pp. 54--83) gives an excellent review.

The role of Lucas sequences in
primality proving was begun by Lucas and cemented by [Morrison75]. Their primitive parts (also known as Sylvester's cyclotomic numbers)
were studied in [Ward1959]. Prime generalized Lucas numbers
are clearly a particular case of prime primitive parts,
occurring when *n* is also a prime. As Ribenboim indicates,
there is an extensive literature on primitive prime Lucas factors,
from [Carmichael1913] to [Voutier1995], via, for example,
[Schinzel1974] and [Stewart1977].

Note: As with many such forms, when the parameters are unrestricted, all primes are of these forms. So in keeping with our definition of generalized repunit primes we require that 5*n* > max(abs(*p*),abs(sqrt(*D*))).

>rank prime digits who when comment 1 U(24, - 25, 43201)60391 CH12 May 2020 Generalized Lucas number 2 U(67, - 1, 26161)47773 x45 Oct 2019 Generalized Lucas number 3 U(2449, - 1, 12671)42939 x45 Nov 2018 Generalized Lucas number, cyclotomy 4 U(21041, - 1, 9059)39159 x45 Nov 2018 Generalized Lucas number, cyclotomy 5 U(5617, - 1, 9539)35763 x45 Jun 2019 Generalized Lucas number, cyclotomy 6 U(1624, - 1, 10169)32646 x45 Nov 2018 Generalized Lucas number, cyclotomy 7 U(2341, - 1, 8819)29712 x25 Apr 2008 Generalized Lucas number 8 U(1404, - 1, 9209)28981 CH10 Nov 2018 Generalized Lucas number, cyclotomy 9 U(2325, - 1, 7561)25451 x20 Oct 2013 Generalized Lucas number 10 U(13084, - 13085, 6151)25319 x45 Nov 2018 Generalized Lucas number, cyclotomy 11 U(1064, - 1065, 8311)25158 CH10 Nov 2018 Generalized Lucas number, cyclotomy 12 (2^{83339}+ 1)/325088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff 13 U(19258, - 1, 5039)21586 x23 Apr 2007 Generalized Lucas number 14 U(11200, - 1, 5039)20400 x25 Mar 2004 Generalized Lucas number, cyclotomy 15 U(8454, - 1, 5039)19785 x25 Jan 2013 Generalized Lucas number 16 U(6584, - 1, 5039)19238 x23 Apr 2007 Generalized Lucas number 17 U(5768, - 5769, 4591)17264 x45 Nov 2018 Generalized Lucas number, cyclotomy 18 U(11091, - 1, 4049)16375 CH3 Sep 2005 Generalized Lucas number 19 U(2554, - 1, 4751)16185 CH3 Oct 2005 Generalized Lucas number 20 U(1599, - 1, 5039)16141 x23 Apr 2007 Generalized Lucas number

- Carmichael1913
R. D. Carmichael, "On the numerical factors of the arithmetic forms α^{n}± β^{n},"Ann. Math.,15(1913) 30--70.- Lucas1878
E. Lucas, "Theorie des fonctions numeriques simplement periodiques,"Amer. J. Math.,1(1878) 184--240 and 289--231.- Morrison75
M. Morrison, "A note on primality testing using Lucas sequences,"Math. Comp.,29(1975) 181--182.MR 51:5469- Ribenboim95
P. Ribenboim,The new book of prime number records, 3rd edition, Springer-Verlag, 1995. New York, NY, pp. xxiv+541, ISBN 0-387-94457-5.MR 96k:11112[An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]- Schinzel1974
A. Schinzel, "Primitive divisors of the expressionA^{n}- B^{n}in algebraic number fields,"J. Reine Angew. Math.,268/269(1974) 27--33.MR 49:8961- Stewart1977
C. L. Stewart, "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers,"Proc. Lond. Math. Soc.,35:3 (1977) 425--447.MR 58:10694- Voutier1995
Voutier, P. M., "Primitive divisors of Lucas and Lehmer sequences,"Math. Comp.,64:210 (1995) 869--888.MR1284673(Annotation available)- Ward1959
M. Ward, "Tests for primality based on Sylvester's cyclotomic numbers,"Pacific J. Math.,9(1959) 1269--1272.MR 21:7180

Chris K. Caldwell
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