The Top Twenty--a Prime Page Collection

Generalized Lucas Number

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(up) Definitions and Notes

In a problem in his text Liber Abbaci (published in 1202), Fibonacci introduced his now famous sequence:
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 {U0, U1, U2, ...} and {V0, V1, V2, ...} for which the next term is P times the current term minus Q times the previous one:
Un+1 = P*Un - Q*Un-1     and     Vn+1 = P*Vn - Q*Vn-1.
We usually require that P and Q be non-zero integers and that (the discriminant) D = P2-4Q is also not zero. 

To define the Lucas sequences, let a and b be the zeros of the polynomial x2-Px+Q, then define the two companion sequences as follows:

Un(P,Q) = (an - bn)/(a - b),     and     Vn(P,Q) = an + bn.
So U0=0, U1=1,V0=2, and V1=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.

These sequences have many useful properties such as: U2n=UnVn; and if p is and odd prime, then p divides Up-(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].

(up) Record Primes of this Type

rankprime digitswhowhencomment
1U(2341, - 1, 8819) 29712 x25 Apr 2008 Generalized Lucas number
2U(2325, - 1, 7561) 25451 x20 Oct 2013 Generalized Lucas number
3(283339 + 1)/3 25088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff
4U(19258, - 1, 5039) 21586 x23 Apr 2007 Generalized Lucas number
5U(11200, - 1, 5039) 20400 x25 Mar 2004 Generalized Lucas number, cyclotomy
6U(8454, - 1, 5039) 19785 x25 Jan 2013 Generalized Lucas number
7U(6584, - 1, 5039) 19238 x23 Apr 2007 Generalized Lucas number
8U(11091, - 1, 4049) 16375 CH3 Sep 2005 Generalized Lucas number
9U(2554, - 1, 4751) 16185 CH3 Oct 2005 Generalized Lucas number
10U(1599, - 1, 5039) 16141 x23 Apr 2007 Generalized Lucas number
11U(14257, - 1, 3779) 15694 x25 Jan 2004 Generalized Lucas number, cyclotomy
12U(551, - 1, 5669) 15537 x25 May 2016 Generalized Lucas number
13U(1493, - 1, 4621) 14665 CH3 Oct 2005 Generalized Lucas number
14U(12924, - 12925, 3499) 14382 x25 Feb 2005 Generalized Lucas number
15U(12113, - 1, 3499) 14284 CH3 Sep 2005 Generalized Lucas number
16U(2441, - 1, 4201) 14228 CH3 Oct 2005 Generalized Lucas number
17U(11194, - 11195, 3361) 13605 x25 Jan 2004 Generalized Lucas number
18U(2219, - 1, 4049) 13546 CH3 Oct 2005 Generalized Lucas number
19U(475, - 1, 5039) 13486 x25 Dec 2003 Generalized Lucas number, cyclotomy
20U(7537, - 7538, 3361) 13028 x23 Mar 2007 Generalized Lucas number

(up) References

Carmichael1913
R. D. Carmichael, "On the numerical factors of the arithmetic forms αn ± βn," 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 expression An - Bn 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
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