Lucas prime
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: 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 are 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, and 51169.
These have been tested by Dubner and Keller to n=50000 [DK99]. Broadhurst and de Water proved v51169 prime. In addition to these provable primes, a number of probable-primes vn have been discovered:
n = 56003, 81671, 89849 [Dubner]; 94823 [H. Lifchitz]; 140057, 148091 [de Water]; 159521, 183089, 193201, 202667 [H. Lifchitz] and 344293, 387433, 443609 [R. Lifchitz].
Renaud Lifchitz has now checked these up to n=400,000.

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.

This page contributed by T. D. Noe.

See Also: FibonacciNumber

References:

BMS1988
J. Brillhart, P. Montgomery and R. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260.  MR 89h:11002
BMS88
J. Brillhart, P. L. Montgomery and R. D. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260, S1--S15.  MR 89h:11002 [See also [DK99].]
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.]



Chris K. Caldwell © 1999-2014 (all rights reserved)