Fermat Number Record

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):

Don't attempt directly to open the worksheet (by clicking on my link), instead right click and save the worksheet to (say) the download folder.

Now that the worksheet is in the download folder (cut it out if you wish and put into whatever folder you wish) you may open the worksheet in the usual way (I should add that while I see this 'works', I have absolutely no idea as to why it does... ) [END OF NOTE].

Announcement (July 1999) of the discovery of the largest known composite Fermat number

382447.gif (516 bytes)

My Maple-based 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.

developments in later years

The above number was found, and proved composite, using Yves Gallot's remarkable Proth program. The discovery was made on one of my College's 350 MHz Pentium computers, as a result of finding the prime number p = 3 times 2³⁸²⁴⁴⁹ + 1. That prime number (at that time the 10^th largest known prime, having 115,130 digits) was found at about 5.00 P.M on Friday 23^rd July 1999, and the extra day-long computation then established that it is a factor of F_382,447.

F_382,447 is also

the first composite Fermat number for which a proper factor has been found having more than 100,000 digits.

the first composite Fermat number with more than 10 ^100,000 decimal digits (more precise details below).

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 10²⁴⁰⁰⁰ times greater than F_303,088. The above Gallot-Proth 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.

Any member of the team could have made this discovery, but few of us could have done it (and certainly not I) without Yves Gallot's initiative and his enormous talent. Complete details concerning the current state of knowledge concerning prime factors of Fermat numbers are available at this location. Chris Caldwell's encyclopaedic Prime Pages site is a good site for prime numbers in general.

The enlightened French Embassy in Dublin sponsored a visit to Ireland by Yves Gallot in October 1999. Yves Gallot was the guest of honour at a reception at my College on Thursday, 28^th October, hosted by my College's President (historian Dr. Pauric Travers), and there followed a public lecture (using Maple), given by me, with the title The history of Fermat numbers from August 1640 . On the following afternoon - Friday 29^th October, at 4.30 P.M, in the Salmon Lecture Theatre of Trinity College, Dublin - Yves Gallot gave a talk on the history of his remarkable program.

(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*2ⁿ + 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 Gallot-Proth 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 2³⁸²⁴⁴⁹ + 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.

(Could I risk going into College at such a time, when we would be about to go out for a meal together? It had to be 'No!', I could come in first thing on Monday morning...) (Did I sleep that night, and the following one? Or on any night since?) On Sunday, just after midday, I checked my home e-mail, and found I had an excited message from Yves Gallot, who, nevertheless, gave nothing away! It would not be proper for me to quote it here, but he advised me that I ought to go back to my College's computer and take a look at its screen. He mentioned four well known people to whom he had already written, but said he would leave it to me to inform... . To say that I was excited would be to put it mildly, but at the same time I didn't wish to inflict a visit on my wife to my College at such a time.

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 F₅ = 2^(2⁵) + 1 = 2³² + 1 = 641 times 6700417) and Fermat.

(And think also of the more modestly talented Landry, who at 82 years of age found that

F₆ = 2^(2⁶) + 1 = 2⁶⁴ + 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 e-mail, and immediately set about checking the computations on his own computer. By mid-morning, 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
in Room D 318 of St. Patrick's College

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 Gallot-Proth computation was lost. However, in Notepad, the entire log of the computations was saved, and today my ever-helpful 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 2³⁸²⁴⁴⁹ + 1, but one can see most of the relevant outputs. One can see the important 'a = 5' and 'a = 11,' and the 'F382447.'

fermat.jpg (20321 bytes)

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:

	If one were to write out the actual base 10 value of the number F_382,447 then it would have approximately 10^115,127.4974 digits!!
	If one were to write out the actual base 10 value of the number F_382,447 in a square grid - at the rate of four digits per inch in the horizontal and vertical direction - then the square would have side length measuring approximately 10^57,545.58733 light years!! [Recall that a light year is the distance travelled by light in one year, and that the speed of light is approximately 186,000 miles per second.] I have prepared a Maple worksheet in which I establish these entirely elementary computations, and it may be downloaded here. (But bear in mind my note at the top of this page.) A text-only html version - for people who do not have (the remarkable) Maple software - is here. Actually a simple mental calculation will establish the above orders of magnitudes. The above numbers might appear to be a bit fantastic (in the sense of being in the realms of fantasy), but they are not, and I would like to treat the first of them with a back-of-an-envelope analysis, which easily establishes the order of magnitude for the number of digits in the base 10 representation of the Fermat number F_382,447. This is how it can be done (a series of simple lower estimates, which is deliberately pitched at school level), merely by using the simple observation that 2¹⁰ = 1024 > 1000 = 10³. My reader should be able to see the appropriate justification at each stage (read left to right along each line, and then continue into the next line. The expressions have had to be created as gif files, and so some look a little unusual, but should not cause any real difficulties. In the 5^thof the expressions, the ' . ' (between the '12' and the '2') is ' times ', and not a ' decimal point '): > = > = > , and thus: > ... (i) Then, since 2^382,445 = (2¹⁰)^38,244.5 > (10³)^38,244.5 = 10^114,733.5 , we finally have, from (i), that > > and so it is immediately clear that F_382,447 has at least 10 ^114,733 digits when expressed in the base 10.

Contact details. jbcosgrave at gmail.com