The first few prime Lucas numbers 
are 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, ... (OEIS
A005479), corresponding to indices 
, 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, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609,
532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849, ... (Dubner
and Keller 1999, Lifchitz and Lifchitz; OEIS A001606).
Only those up to index 56003 have been proven prime (Broadhurst and Irvine 2006;
http://primes.utm.edu/primes/page.php?id=77992).
As of Apr. 2009, the largest known Lucas probable
prime is 
,
which has 
decimal digits (R. Lifchitz, Mar. 2009).
Lucas Prime
See also
Fibonacci Prime, Integer Sequence Primes, Lucas Number, Lucas PseudoprimeExplore with Wolfram|Alpha
References
Brillhart, J.; Montgomery, P. L.; and Silverman, R. D. "Tables of Fibonacci and Lucas Factorizations." Math. Comput. 50, 251-260, 1988.Dubner, H. and Keller, W. "New Fibonacci and Lucas Primes." Math. Comput. 68, 417-427 and S1-S12, 1999.Lifchitz, H. and Lifchitz, R. "PRP Top Records." http://www.primenumbers.net/prptop/searchform.php?form=L(n).Referenced on Wolfram|Alpha
Lucas PrimeCite this as:
Weisstein, Eric W. "Lucas Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LucasPrime.html