Lucas Prime

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; https://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).

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." https://www.primenumbers.net/prptop/searchform.php?form=L(n).