## Lucas Number |

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:

v_{1}= 1,v_{2}= 3 andv_{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

(See the list below for any larger ones found more recently.) These were tested by Dubner and Keller ton= 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.

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.

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 Dec 1995 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

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

- BMS1988
J. Brillhart,P. MontgomeryandR. 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. DubnerandW. Keller, "New Fibonacci and Lucas primes,"Math. Comp.,68:225 (1999) 417--427, S1--S12.MR 99c:11008[Probable primality ofF,L,F*andL*tested fornup to 50000, 50000, 20000, and 15000, respectively. Many new primes and algebraic factorizations found.]

Chris K. Caldwell
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