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<caldwell@utm.edu> 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 and v_{n+1} = v_{n} + v_{n1} (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 v_{51169} prime. In addition to these provable primes, a number of probableprimes 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.
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Chris K. Caldwell © 19992014 (all rights reserved)
