 
New Composite Fermat Number Records
in February and October
2003
On
Monday 17^{th }February
2003, Thomas Walsh (one of my college ProthGallot
Group)  [1] here is
Thomas eight days later, with Jacqueline Fallon, another member of my
computation group, and here
[2]
Thomas points at his results file  found a George Woltman PRPdiscovered
645817digit probable prime p = (3X2^{2145353} +
1) [3] on his
computer (I had already done sieve work with Paul Jobling's NewPGen).
That evening, at home, I began using Yves Gallot's Proth
program to check for the (almost certain) primality; that would normally
entail two independent Proth tests, followed by the tense period of Fermat and generalised Fermat number testing. By Wednesday 19^{th }
(after 40 hours, the first
round was over: [4] a definite
prime), but later that day Yves sent me a modified program which would allow
me to split the nervejangling sequence of calculations: some at home and some
here in my office.
At exactly 14:56:24, at home, on Friday 21^{st
} I saw (in a
heartstopping moment; actually for several seconds I couldn't bring myself to
look...) that (3X2^{214535}^{3} +
1) [5] divided the
2145351^{th} Fermat number, breaking the
previous
record set in July 1999 by Yves Gallot and me. (I don't have a digital camera, so I transferred my
home Proth screen to college, by saving the .dta etc files to a floppy, and
temporarily reran a second version of Proth here at my desk).
Shortly afterwards I sent off an official
notification to the Number Theory Mailing List. Wilfrid Keller's outstanding
Fermat number site (which will inform you of who did what, and when,
right back to
Euler's
finding of a factor of the fifth Fermat number in 1732) is here,
and there is simply nowhere better to look for information on primes in general than Chris Caldwell's leading prime
number site.
Wilfrid Keller (WK) and
Chris Caldwell (CC) maintain
current details concerning generalised Fermat numbers for the 'historic'
bases (6, 10 and 12): WK6,
WK10, WK12,
CC6, CC10,
CC12. Now
(March) I
have reported that the above prime attains a new record for all the
historic bases: 3, 5, 6, 10 and 12. These three photos show the final
ProthFermat screens for all the new records: [6]
two screens, [7]
Proth prime and GFNs, and [8] GFNs.
My thanks to Paul Murphy for all the above digital
photos.
Ivars Peterson devoted his Science News MathTrek article
of 1^{st} March 2003 to the
above discovery.
On
Tuesday 7^{th }October
2003 I noticed the 746190digit probable prime on the computer of M�ire N� Bhaoill (M�ire is one of my St Patrick's College computational group).
I had done, as always, the preliminary sieve work with Paul Jobling's newpgen program, and the probabilistic primality testing with George
Woltman's PRP program. By Friday 10^{th}
(3X2^{247878}^{5} + 1) had been established to be a deterministic prime (using Yves Gallot's Proth program), and to be a factor of the 2478782^{nd} Fermat number
(screen photo), using Yves Gallot's GFN program.
I sent notification
within hours to the Number Theory Mailing List.
The size of that Fermat number
almost defies belief: to write it out in decimal notation (i.e. in the
normal fashion), at, say 4 digits per inch in the horizontal and vertical, would
require a square whose side exceeded 10^{373075}
LIGHT YEARS... (interested readers may read my elementary analysis of that topic
in the
1999 Fermat Record
Number corner of this site). The above October discovery breaks the previous
record announced in February 2003.
� 


