New Fibonacci and Lucas primes

Abstract:

Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers $F_{n}$ have been determined for $6000 < n \le 50000$ and all probable prime Lucas numbers $L_{n}$ have been determined for $1000 < n \le 50000$. A rigorous proof of primality is given for $F_{9311}$ and for numbers $L_{n}$ with $n = 1097$, $1361$, $4787$, $4793$, $5851$, $7741$, $10691$, $14449$, the prime $L_{14449}$ having 3020 digits. Primitive parts $F^{*}_{n}$ and $L^{*}_{n}$ of composite numbers $F_{n}$ and $L_{n}$ 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 $F_{n}$ and $L_{n}$ are given for $n > 1000$ as far as they have been completed, adding information to existing factor tables covering $n \le 1000$.