Date: Wed, 1 Feb 1995 18:14:31 EST Reply-To: Harvey Dubner <70372.1170@compuserve.com> Sender: Number Theory List From: Harvey Dubner <70372.1170@compuserve.com> Subject: Fibonacci/Lucas primes To: Multiple recipients of list NMBRTHRY The following abstract appeared in a recent "Abstract of Papers presented at the AMS." ------------------------------------------------ HARVEY DUBNER, Beverly Road, Ridgewood, New Jersey 07450, and WILFRID KELLER, Regionales Rechenzentrum der Universit"at Hamburg, 20146 Hamburg, Germany. New Fibonacci and Lucas primes. Preliminary report. Fibonacci numbers F_n and the related Lucas numbers L_n are defined recursively by the formulas F_{n+2} = F_{n+1} + F_n and L_{n+2} = L_{n+1} + L_n, where F_0 = 0, F_1 = 1 and L_0 = 2, L_1 = 1, respectively. Extending previous work of J. Brillhart and H. C. Williams [as reported in Math. Comp. 50 (1988), p. 255], all primes F_n for 2971 < n < 9677 and all primes L_n for 863 < n < 8467 have been determined. Thus, F_n proved to be prime for n = 4723, 5387, 9311, and L_n proved to be prime for n = 1097, 1361, 4787, 4793, 5851, 7741. The Fibonacci numbers F_{4723} and F_{5387} had been given as probable primes by Williams. Rigorous proofs of primality became possible due to the multiplicative structure of F_n +- 1 and L_n +- 1. Beyond the mentioned intervals, the first-named author showed that numbers F_n are probable primes for n = 9677, 14431, 25561, 30757, 35999, 37511, and for no other n <= 50000, and that numbers L_n are probable primes for n = 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, and for no other n <= 50000. The primality proof for the 3020-digit Lucas number L_{14449} has also been completed. --------------------------------------------- A paper is being prepared. It is interesting that this is the first time that the 1000-digit wall has been broken for Fibonacci/Lucas numbers. That is, the following have joined the ranks of Titanic primes: F_5387 1226 digits F_9311 1946 digits L_4787 1001 digits L_4793 1002 digits L_5851 1223 digits L_7741 1618 digits L_14449 3020 digits I am happy to report that the Dubner Cruncher played a significant role in this project. If anyone wants any detailed information about the Cruncher, just ask. Harvey Dubner