Record from John B. Cosgrave
Largest non-Mersene prime being a divisor Fermat number.

----- Original Message -----
From: "John B. Cosgrave"
To: NMBRTHRY@LISTSERV.NODAK.EDU
Sent: Saturday, February 22, 2003 3:00 AM
Subject: a new largest known composite Fermat number


I am absolutely delighted to announce a new largest composite Fermat number:

F[2145351] = 2^(2^2145351) + 1 is divisible by the 645817-digit prime
p = (3*2^2145353 + 1)

(In passing I note that p is now the fifth largest known, and the
largest known non-Mersenne prime).

This work was done using Paul Jobling's 'newpgen', George Woltman's
'PRP', and Yves Gallot's 'proth' programs. Besides their
encouragement, and that of family, friends and my college colleagues,
I would wish to especially acknowledge Wilfrid Keller. I would like
also to express my admiration for Manfred Toplic who has been working
for a long time with (5*2^n + 1); I hope that one day soon he will be
rewarded with a factor of a large Fermat number.

The previous largest known composite Fermat number - found by Yves
Gallot and me in July 1999 - had been F[382447], divisible by the
115130-digit prime q = (3*2^382449 + 1). Until now q had been the only
known proper factor of a Fermat number with more than 100000
digits. Details ([1]).

After July 1999 I continued searching for other primes of the form
(3*2^n + 1), and in October 2000 I founded the Proth-Gallot group at
my college ([2]), and passed the n = 1000000 mark by July 2001, using
as many college student machines as possible during holiday
periods. In May 2001 I found the prime 275977-digit prime (3*2^916773
+ 1); it didn't divide a Fermat number, but did divide a number of
generalised ones ([3]). I reached n = 2000000 by November 2002 without
finding another prime.

On Monday 17th February 2003 I found PRP had recorded (3*2^2145353 +
1) as a probable prime on Thomas Walsh's computer (Thomas is one of my
college group):

[Sun Feb 16 08:41:55 2003] 3*2^2145353+1 is a probable prime.

and later that day I subjected it to Yves Gallot's 'proth' program,
which confirmed it was prime (as expected). Here is today's proth.log:
Wed Feb 19 20:10:51 2003 : Start: For n=2145353 to 2145353, For k=3 to
3 step 2, k*2^n+1. Fri Feb 21 14:56:24 2003 : 3*2^2145353 + 1 divides
GF(2145351, 2) !

I will report other (expected) GF divisibility results at my web site
in the coming week in a to-be-created corner (with one digital photo,
taken in college on Monday of Thomas Walsh's screen showing PRP
'probable' prime).

Currently I am PRP-ing to 2280000, and for some time have been doing
longterm 'newpgen' preparatory work out to 4000000.

Finally (and I realise this is silly, but I got asked this sort of
question in July 1999 and later) the size of F[2145351] is truly
awesome: to write out its decimal value - at 4 digits per inch in the
horizontal and vertical directions - would require a square sheet of
paper with side length exceeding 10^322889 light years (use my Maple
worksheet how-many.mws from [3]).

John Cosgrave

[1] http://www.spd.dcu.ie/johnbcos/fermat.htm


[2] http://www.spd.dcu.ie/johnbcos/proth-gallot_group_(spd).htm


[3] http://www.spd.dcu.ie/johnbcos/916773.htm


John B. Cosgrave, Mathematics Department, St. Patrick's College,
Drumcondra, Dublin 9, IRELAND.
Home email: johnbcos@iol.ie
College email: John.Cosgrave@spd.dcu.ie

Royalties from my *A Prime For The Millennium*, from Tim and
Mairead Robinson's Folding Landscapes
                    http://www.iol.ie/~tandmfl
are being donated to the Irish Cancer Society.

F[382447], the largest known composite Fermat number, was
discovered by Yves Gallot and me in July 1999. Unimaginably
large, its decimal digits cannot be written out in the entire
universe. Details at my college web site:
                    http://www.spd.dcu.ie/johnbcos