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."
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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