## Generalized Lucas Number |

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

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 (2^{95369}+ 1)/328709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP 10 U(2325, - 1, 7561)25451 x20 Oct 2013 Generalized Lucas number 11 U(13084, - 13085, 6151)25319 x45 Nov 2018 Generalized Lucas number, cyclotomy 12 U(1064, - 1065, 8311)25158 CH10 Nov 2018 Generalized Lucas number, cyclotomy 13 (2^{83339}+ 1)/325088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff 14 U(19258, - 1, 5039)21586 x23 Apr 2007 Generalized Lucas number 15 U(11200, - 1, 5039)20400 x25 Mar 2004 Generalized Lucas number, cyclotomy 16 U(8454, - 1, 5039)19785 x25 Jan 2013 Generalized Lucas number 17 U(6584, - 1, 5039)19238 x23 Apr 2007 Generalized Lucas number 18 U(5768, - 5769, 4591)17264 x45 Nov 2018 Generalized Lucas number, cyclotomy 19 U(11091, - 1, 4049)16375 CH3 Sep 2005 Generalized Lucas number 20 U(2554, - 1, 4751)16185 CH3 Oct 2005 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, New York, NY, 1995. 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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