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.

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:

U_n+1 = P*U_n - Q*U_n-1 and V_n+1 = P*V_n - Q*V_n-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:

U_n(P,Q) = (aⁿ - bⁿ)/(a - b), and V_n(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: U_2n=U_nV_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 5n > max(abs(p),abs(sqrt(D))).

Record Primes of this Type

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 (2¹³⁸⁹³⁷ + 1)/3 41824 E12 Oct 2023 Wagstaff, ECPP, generalized Lucas number
5 U(21041, - 1, 9059) 39159 x45 Nov 2018 Generalized Lucas number, cyclotomy
6 (2¹²⁷⁰³¹ + 1)/3 38240 E5 Jan 2023 Wagstaff, ECPP, generalized Lucas number
7 U(5617, - 1, 9539) 35763 x45 Jun 2019 Generalized Lucas number, cyclotomy
8 (2¹¹⁷²³⁹ + 1)/3 35292 E2 Aug 2022 Wagstaff, ECPP, generalized Lucas number
9 U(1624, - 1, 10169) 32646 x45 Nov 2018 Generalized Lucas number, cyclotomy
10 U(2341, - 1, 8819) 29712 x25 Apr 2008 Generalized Lucas number
11 U(1404, - 1, 9209) 28981 CH10 Nov 2018 Generalized Lucas number, cyclotomy
12 (2⁹⁵³⁶⁹ + 1)/3 28709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP
13 U(2325, - 1, 7561) 25451 x20 Oct 2013 Generalized Lucas number
14 U(13084, - 13085, 6151) 25319 x45 Nov 2018 Generalized Lucas number, cyclotomy
15 U(1064, - 1065, 8311) 25158 CH10 Nov 2018 Generalized Lucas number, cyclotomy
16 (2⁸³³³⁹ + 1)/3 25088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff
17 U(19258, - 1, 5039) 21586 x23 Apr 2007 Generalized Lucas number
18 U(11200, - 1, 5039) 20400 x25 Mar 2004 Generalized Lucas number, cyclotomy
19 U(8454, - 1, 5039) 19785 x25 Jan 2013 Generalized Lucas number
20 U(6584, - 1, 5039) 19238 x23 Apr 2007 Generalized Lucas number

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