The Top Twenty--a Prime Page Collection

Generalized Lucas Number

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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.

(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].

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 5n > max(abs(p),abs(sqrt(D))).

(up) Record Primes of this Type

rankprime digitswhowhencomment
1U(24, - 25, 43201) 60391 CH12 May 2020 Generalized Lucas number
2U(67, - 1, 26161) 47773 x45 Oct 2019 Generalized Lucas number
3U(2449, - 1, 12671) 42939 x45 Nov 2018 Generalized Lucas number, cyclotomy
4U(21041, - 1, 9059) 39159 x45 Nov 2018 Generalized Lucas number, cyclotomy
5U(5617, - 1, 9539) 35763 x45 Jun 2019 Generalized Lucas number, cyclotomy
6U(1624, - 1, 10169) 32646 x45 Nov 2018 Generalized Lucas number, cyclotomy
7U(2341, - 1, 8819) 29712 x25 Apr 2008 Generalized Lucas number
8U(1404, - 1, 9209) 28981 CH10 Nov 2018 Generalized Lucas number, cyclotomy
9(295369 + 1)/3 28709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP
10U(2325, - 1, 7561) 25451 x20 Oct 2013 Generalized Lucas number
11U(13084, - 13085, 6151) 25319 x45 Nov 2018 Generalized Lucas number, cyclotomy
12U(1064, - 1065, 8311) 25158 CH10 Nov 2018 Generalized Lucas number, cyclotomy
13(283339 + 1)/3 25088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff
14U(19258, - 1, 5039) 21586 x23 Apr 2007 Generalized Lucas number
15U(11200, - 1, 5039) 20400 x25 Mar 2004 Generalized Lucas number, cyclotomy
16U(8454, - 1, 5039) 19785 x25 Jan 2013 Generalized Lucas number
17U(6584, - 1, 5039) 19238 x23 Apr 2007 Generalized Lucas number
18U(5768, - 5769, 4591) 17264 x45 Nov 2018 Generalized Lucas number, cyclotomy
19U(11091, - 1, 4049) 16375 CH3 Sep 2005 Generalized Lucas number
20U(2554, - 1, 4751) 16185 CH3 Oct 2005 Generalized Lucas number

(up) References

R. D. Carmichael, "On the numerical factors of the arithmetic forms αn ± βn," Ann. Math., 15 (1913) 30--70.
E. Lucas, "Theorie des fonctions numeriques simplement periodiques," Amer. J. Math., 1 (1878) 184--240 and 289--231.
M. Morrison, "A note on primality testing using Lucas sequences," Math. Comp., 29 (1975) 181--182.  MR 51:5469
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.]
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
C. L. Stewart, "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers," Proc. Lond. Math. Soc., 35:3 (1977) 425--447.  MR 58:10694
Voutier, P. M., "Primitive divisors of Lucas and Lehmer sequences," Math. Comp., 64:210 (1995) 869--888.  MR1284673 (Annotation available)
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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