
A note of late March 2017 concerning the opening of the Maple worksheets below, indeed my Maple worksheets
in general.
You may encounter a problem opening my Maple worksheets (as indeed I do myself) here at my web site  it would appear to depend on the internet browser being used.
Thus, if I attempt to open one of my worksheets using Interner Explorer there is never a problem, whereas if I use Firefox then all that one sees
 this is just an example  is something like this:
{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" 1 0 "Courier" 0 0 128 0 128 1 0 1 0
0 1 0 0 0 0 }{CSTYLE "2D Math" 1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0
0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "2D Output" 2 20 "" 0 0 0 128 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "
" 1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 1 257 "" 0 1 0
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0
0 0 0 }{CSTYLE "" 1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE ""
and so on (almost at infinitum)... I asked Maplesoft for advice on this and they recommended doing the following (and I found it worked):
Announcement (July 1999) of the discovery of the largest known composite Fermat number
My Maplebased public lecture (in mws and html format)  The history of Fermat numbers from August 1640  may be accessed in the Public and other lectures section of my site. A related paper, written by Yves Gallot, may be accessed here.
The previous record composite Fermat number:
was announced by Jeffrey Young in
Mathematics of Computation  an
American Mathematical Society journal  in early 1998.
Jeffrey Young found that F_{303,088}  having just over 10 ^{90,000} decimal digits  has the 91,241 decimal digits prime factor 3*2^{303,093} + 1.
The new record composite Fermat number  F_{382,447}  is almost
10^{24000}
times greater than F_{303,088}.
The above GallotProth discovery is the fruit of an international collaborative effort led by
Yves Gallot
 the brilliant creator of the Proth.exe program  ably helped, in the compilation of the work, by the valiant Ray Ballinger.
(In contrast to the enlightened French Embassy I make this aside comment about the United States Embassy here in Dublin. In 1999 Richard Crandall, Ernst Mayer, and Jason Papadopoulos proved that the 24th Fermat number (which has 5050446 decimal digits) is composite (i.e., is not a prime number); they did so by applying Pépin's test (explained here at this Wikipedia page; I used to teach this material to my students). Their AMS paper explaining this discovery is available here. I was fortunate to be on friendly terms with Richard and wrote to ask if he would come to Dublin to talk about this providing I could get some funding from a USA source, ideally his country's Embassy here in Dublin... . Yes, Richard was willing, and I tried, and I tried, ... , but the USA Embassy wouldn't put up a cent... . I tried putting pressure by mentioning the enlightened French Embassy, but to no avail... ) An informal history of the discovery. For some weeks I had been checking for primality numbers of the form 3*2^{ n } + 1 with n in the range 366,000 to 390,000. Then, just before 5.00 P.M. on the afternoon of Friday, 23^{rd} July 1999, I left my office to check on the state of the GallotProth computations in College's main computer lab., Room D 318. I refreshed the screens of computer after computer, and clicked up the screens to view earlier computations.
In the third row, I examined computer #17, the second last one in that row. After a few seconds I registered  with great delight  that the number 3 times 2^{382449} + 1 was a prime (and had only been registered as such some minutes earlier). I saw that there was a consequent computation in progress  the one that would test for possible division into related Fermat numbers  which was going to take some "1200mn 27sec" to complete. Thus the outcome would be known at about 5.00 P.M. the following day, the 28^{th} wedding anniversary of myself and my wife, Mary.
I went downstairs to tell my wife that we had almost certainly found the largest composite Fermat number, otherwise Yves Gallot's letter simply would not be sensible... After lunch, and setting off for an afternoon's walk to Dalkey, I began to give my wife yet another lecture on the beauty of Fermat numbers, and  possibly to shut me up, though actually because  as she said  I would get no sleep that night, she suggested we go back home, get our bikes, and cycle over to College...
On the way I could think of nothing else but my boyhood mathematical heroes Euler (who was the first to find a composite Fermat number, the hand sized F_{5}
=
2^{(25)} + 1 = 2^{32} + 1 = 641 times 6700417)
and Fermat.
F_{6} = 2^{(26)} + 1 = 2^{64} + 1 = 274177 times 67280421310721) By the time we got up to Room D 318 I was in quite a state (only some hours later there was a power failure in College; if that had happened before we had arrived...). To computer #17, a touch to the mouse to refresh the screen, a couple of clicks up the side to get back to the earlier outputs, and there it was!! What I learned from correspondence with Yves Gallot. Yves Gallot was about to leave his office in Toulouse that Friday afternoon when he checked at Chris Caldwell's site to see which new large primes had been submitted that day, and he saw the one I had submitted a few hours earlier. On his arrival home he also saw my email, and immediately set about checking the computations on his own computer. By midmorning, Sunday 25th he knew that his program had discovered the largest known composite Fermat number... . What joy. After recovering a little, he emailed me, suggesting that I have a look at my College computer...
Digital photograph of the screen of computer #17 College suffered a power failure sometime in the evening of Sunday 25^{th} of July, as a result of which the full screen version of the GallotProth computation was lost. However, in Notepad, the entire log of the computations was saved, and today my everhelpful colleague Paul Murphy took a number of digital photos of the screen of computer #17 (appropriately '17' is a Fermat number!!). Unfortunately you cannot see the initial part of the 3 times 2^{382449} + 1, but one can see most of the relevant outputs. One can see the important 'a = 5' and 'a = 11,' and the 'F382447.'
How large is the number F_{382,447}? I was asked that question many, many times in the days following the discovery. My initial response was something along the lines of: Oh! It is utterly gigantic, astronomically large, etc., etc. Then I decided to stop giving vague answers, and give a fairly detailed response (which may not be all that more helpful), and it is this:
