# 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.

### Definitions and Notes

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, as of June 2017, were 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, 51169, 56003, 81671, 89849, 94823, 140057 and 148091.
(See the list below for any larger ones found more recently.) These were tested by Dubner and Keller to n=50000 last century [DK99]. In addition to these provable primes, a number of probable-primes vn have been discovered, see the Lifchitz's PRP site linked below for a list.

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.

### Record Primes of this Type

rankprime digitswhowhencomment
1V(148091) 30950 c81 Sep 2015 Lucas number, ECPP
2V(140057) 29271 c76 Dec 2014 Lucas number, ECPP
3V(94823) 19817 c73 May 2014 Lucas number, ECPP
4V(89849) 18778 c70 Jan 2014 Lucas number, ECPP
5V(81671) 17069 c66 Sep 2013 Lucas number, ECPP
6V(56003) 11704 p193 Jun 2006 Lucas number
7V(51169) 10694 p54 Apr 2001 Lucas number
8V(44507) 9302 CH3 Sep 2005 Lucas number
9V(36779) 7687 CH3 Sep 2005 Lucas number
10V(35449) 7409 p12 Mar 2001 Lucas number
11V(19469) 4069 x25 Jan 2002 Lucas number, cyclotomy, APR - CL assisted
12V(14449) 3020 DK Mar 1995 Lucas number
13V(13963) 2919 c11 Jan 2002 Lucas number, ECPP
14V(12251) 2561 p54 May 2001 Lucas number
15V(10691) 2235 DK Jan 1996 Lucas number
16V(8467) 1770 c2 Oct 2000 Lucas number, ECPP
17V(7741) 1618 DK Mar 1995 Lucas number
18V(5851) 1223 DK Mar 1995 Lucas number
19V(4793) 1002 DK Mar 1995 Lucas number
20V(4787) 1001 DK Mar 1995 Lucas number

### 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.]