# Lucas prime

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!

First of the Lucas primes are *v*_{n} with

Larger ones will be added to the Top 20's page on Lucas numbers when found.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, 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.

This page contributed by T. D. Noe.

**See Also:** FibonacciNumber

**References:**

- BMS1988
J. Brillhart,P. MontgomeryandR. Silverman, "Tables of Fibonacci and Lucas factorizations,"Math. Comp.,50(1988) 251--260.MR 89h:11002- BMS88
J. Brillhart,P. L. MontgomeryandR. 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. 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.]

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