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

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 v_n+1 = v_n + v_n-1 (n > 2)

It can be shown that, for odd m, v_n divides v_nm.  Hence, for v_n 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 v_n 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 v_n 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

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

Related Pages

• Henri Lifchitz & Renaud Lifchitz's PRP Top Records (searched for L(?))

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.]