The Prime Glossary: Lucas prime

Lucas prime
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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 are 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, 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 v_n 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.]