New Fibonacci and Lucas primes

Abstract

Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers F n F_{n} have been determined for 6000 > n ≤ 50000 6000 > n \le 50000 and all probable prime Lucas numbers L n L_{n} have been determined for 1000 > n ≤ 50000 1000 > n \le 50000 . A rigorous proof of primality is given for F 9311 F_{9311} and for numbers L n L_{n} with n = 1097 n = 1097 , 1361 1361 , 4787 4787 , 4793 4793 , 5851 5851 , 7741 7741 , 10691 10691 , 14449 14449 , the prime L 14449 L_{14449} having 3020 digits. Primitive parts F n ∗ F^{*}_{n} and L n ∗ L^{*}_{n} of composite numbers F n F_{n} and L n L_{n} have also been tested for probable primality. Actual primality has been established for many of them, including 22 with more than 1000 digits. In a Supplement to the paper, factorizations of numbers F n F_{n} and L n L_{n} are given for n > 1000 n > 1000 as far as they have been completed, adding information to existing factor tables covering n ≤ 1000 n \le 1000 .