The Top Twenty--a Prime Page Collection

Lucas Number

This page : Definition(s) | Records | References | RSS 2.0 Feed
  View this page in:   language help

The Prime Pages

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

This page contributed by T. D. Noe.

A Lucas prime is a Lucas number that is prime.  Recall that the Lucas numbers can be defined as follows:

v1 = 1, v2 = 3 and vn+1 = vn + vn-1 (n > 2)

It can be shown that, for odd m, vn divides vnm.  Hence, for vn to be a prime, the subscript n must be a prime, a power of 2, or zero. However, a prime or power of 2 subscript is not sufficient!

The known Lucas primes are vn with

n = 0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, and 51169.
These have been tested by Dubner and Keller to n=50000 [DK99]. Broadhurst and de Water proved v51169 prime. In addition to these provable primes, a number of probable-primes vn have been discovered:
n = 56003, 81671, 89849 [Dubner]; 94823 [H. Lifchitz]; 140057, 148091 [de Water]; 159521, 183089, 193201 and 202667 [H. Lifchitz].

As with the Fibonacci primes and the Mersenne primes, it is conjectured that there are infinitely many Lucas primes.  Interestingly, all three types of numbers are generated by simple recurrence relations.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1V(56003) 11704 p193 Jun 2006 Lucas number
2V(51169) 10694 p54 Apr 2001 Lucas number
3V(44507) 9302 CH3 Sep 2005 Lucas number
4V(36779) 7687 CH3 Sep 2005 Lucas number
5V(35449) 7409 p12 Mar 2001 Lucas number
6V(19469) 4069 x25 Jan 2002 Lucas number, cyclotomy, APR - CL assisted
7V(14449) 3020 DK Mar 1995 Lucas number
8V(13963) 2919 c11 Jan 2002 Lucas number, ECPP
9V(12251) 2561 p54 May 2001 Lucas number
10V(10691) 2235 DK Dec 1995 Lucas number
11V(8467) 1770 c2 Oct 2000 Lucas number, ECPP
12V(7741) 1618 DK Mar 1995 Lucas number
13V(5851) 1223 DK Mar 1995 Lucas number
14V(4793) 1002 DK Mar 1995 Lucas number
15V(4787) 1001 DK Mar 1995 Lucas number

(up) References

BMS1988
J. Brillhart, P. Montgomery and R. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260.  MR 89h:11002
Brillhart1999
J. Brillhart, "Note on Fibonacci primality testing," Fibonacci Quart., 36:3 (1998) 222--228.  MR1627388
DK99
H. Dubner and W. Keller, "New Fibonacci and Lucas primes," Math. Comp., 68:225 (1999) 417--427, S1--S12.  MR 99c:11008 [Probable primality of F, L, F* and L* tested for n up to 50000, 50000, 20000, and 15000, respectively. Many new primes and algebraic factorizations found.]
Chris K. Caldwell © 1996-2013 (all rights reserved)