Date: Tue, 3 Aug 1999 13:27:14 -0400
To: Number Theory List <NMBRTHRY@LISTSERV.NODAK.EDU>
From: John B. Cosgrave <johnbcos@iol.ie>
Subject: announcing the largest known composite Fermat number
ANNOUNCING THE LARGEST KNOWN COMPOSITE FERMAT NUMBER
Dear Friends and colleagues,
Using Yves Gallot's remarkable Proth program I have made the fortuitous
discoveries that:
1. p = (3*2^382449 + 1) is prime (the 10th. largest known one, and the
3rd. largest non-Mersenne prime),
2. p is a divisor of the Fermat number F[382447] = 2^(2^382447) + 1,
making F[382447] be the largest known composite Fermat number, and
the sixth Fermat number for which a factor has been found using
Gallot's program.
3. p is a divisor of the following 'generalized Fermat numbers':
GF[382443, 6] = 6^(2^382443) + 1,
GF[382447, 3] = 3^(2^382447) + 1,
GF[382447, 12] = 12^(2^382447) + 1,
4. p is NOT a divisor of ANY GF[5, m] or GF[10, m],
5. p is a 'generalized Cullen prime.'
Previously the largest known composite Fermat number was F[303088] =
2^(2^303088) + 1, with factor (3*2^303093 + 1) [Young, 1998]
I made my chance discovery while making a systematic Proth-Gallot test of
all numbers (3*2^n + 1), with 'n' ranging between 366,000 and 390,000,
spread over 40-50 machines in my College's main computer laboratory,
during the past two months.
Best wishes to you all,
John
REFERENCES
1. Wilfrid Keller maintains the 'Prime factors k*2^n + 1 of Fermat
numbers F[m] and complete factoring status' site at:
<http://vamri.xray.ufl.edu/proths/fermat.html>
2. Ray Ballinger valiantly maintains the Proth prime search site at:
<http://vamri.xray.ufl.edu/proths/>
3. Chris Caldwell maintains [just in case you didn't know] the
remarkable Prime number site at:
<http://www.utm.edu/research/primes>
######################################################
#
# John B. Cosgrave
# Mathematics Department,
# St. Patrick's College,
# Drumcondra,
# Dublin 9,
# IRELAND.
#
# [St. Patrick's College is a College
# of Dublin City University]
#
# Home e-mail: johnbcos@iol.ie
# College e-mail: John.Cosgrave@spd.ie
# My College Web site: http://www.spd.dcu.ie/johnbcos
#
# Is 2^p - 1 prime for infinitely many p? (I hope so.)
# Is 2^(2^n) + 1 prime for some n > 4? (Surely at least one?)
# Is Pi^e transcendental? (Probably.)
# Is 2^sqrt(2) + 3^sqrt(3) irrational? (Almost certainly.)
# Is zeta(3) a rational multiple of Pi^3? (Hardly.)
#
######################################################
APH