Mersenne Digest Wednesday, November 24 1999 Volume 01 : Number 663 ---------------------------------------------------------------------- Date: Sun, 21 Nov 1999 23:55:16 EST From: [EMAIL PROTECTED] Subject: Mersenne: Re: Mersenne Digest V1 #662 <<Of course, we've got a classic chaotic feedback mechanism in place... if the list is too quiet (posts per unit time is low), a thread will spontaneously create itself to discuss the fact, immediately rendering the list less quiet (increasing the post per unit time).>> Excellent. And don't forget, when the list gets TOO busy, mails will erupt decrying the current thread (poaching, prize division, and the other infamous ones). This temporarily increases the traffic as people not only argue about the big thread but about the size of the thread and whether it should continue, and then the thread kills itself/is killed. <<The normal traffic (e-mails not related to the lack of e-mails) could be chaotic, but over the long haul is inexorably linked (I'd wager) to the distributions of mersenne primes. That is, if you were to align the graphs of mailing list traffic by time next to our progress towards finding mersenne primes by time, you'd see huge influxes of traffic when we found another prime (and probably when someone else finds one). That sort of predictable dependance (fight about how predictable mersenne primes are later) on an outside 'force' makes that sort of traffic non-chaotic... right?>> A driven chaotic system.... right? I have no idea about terminology. Weather is driven by the sun (105 degrees in the middle of December in Montana? I think not.) but is still chaotic. Of course... *evil grin*... far between discoveries of Mersenne primes we should expect posts complaining about the non-discovery of any Mersenne primes! By the way... I hope to never live to see the day when a new non-GIMPS prime is found. I even feel bad about living through the last days of the Era of Supercomputers. :-D The sun will never set on GIMPS! S. "Expect my cool paper on Mersenne primes out on the 23rd, which oughta put some pep into the list, and yes, I do like giving myself long pseudonicknames" L. PostScript: By the way, if I'm right, we should see one heck of a HUGE increase in list traffic when the missing Mersenne prime is found! *grin* _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Sun, 21 Nov 1999 23:55:58 EST From: [EMAIL PROTECTED] Subject: Mersenne: Re: Mersenne Digest V1 #662 << Things tend to come in clusters.... >> That's what the Noll Island Theory says! Hehehe. S.T.L. _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Mon, 22 Nov 1999 07:24:29 +0100 (CET) From: Henrik Olsen <[EMAIL PROTECTED]> Subject: Re: Mersenne: Re: Mersenne Digest V1 #662 On Sun, 21 Nov 1999 [EMAIL PROTECTED] wrote: > << Things tend to come in clusters.... >> > > That's what the Noll Island Theory says! Hehehe. > > S.T.L. Since there's no upper limit to the size of the gap between consequtive primes, all primes come in clusters, for any reasonable[1] definition of clusters. [1] and if it's not enough for _your_ definition of clusters, then I can always call it unreasonable. That's the way to win disputes. :) - -- Henrik Olsen, Dawn Solutions I/S URL=http://www.iaeste.dk/~henrik/ I'd like somebody to sacrifice their first-borne son, and write in blood that gas does the right thing every time these days. Otherwise I will keep the thing that looks strange but has a real explanation for it. Linus Torvalds about buggy looking code that isn't _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Mon, 22 Nov 1999 13:26:03 +0000 From: "Steinar H. Gunderson" <[EMAIL PROTECTED]> Subject: Re: Mersenne: Quiet List.. On Sun, Nov 21, 1999 at 02:23:16AM -0000, Ian L McLoughlin wrote: >p.s My chance of 6 numbers from the 12 from 49 is,er rrr15,134:1 , I >think(No guarantee of course as it would cost me er...�928 every week.. >Well, its still better than 19,283,416, I think er ..of 6 from 49 pool.. All combinations are equally (un)likely. The only thing that actually could increase your average `winnings' (rather decrease your average losses) would be playing numbers that nobody else did. Then the price would be split among fewer, and your winnings (when you really won the top prize) would go up. In Norway, there are 4000 persons every week playing 1-2-3-4-5-6-7 (we draw 7 numbers from 34) -- needless to say, the prize for 1-2-3-4-5-6-7 is 1/4000th of the largest possible prize that week :-) /* Steinar */ _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Mon, 22 Nov 1999 06:47:10 -0500 From: Matthew Smith <[EMAIL PROTECTED]> Subject: Re: Mersenne: Unreservation Okay. I just upgraded to 19.1. Prime95's first action as a newer entity was to unreserve yet another of my exponents. I requested more, so now I have a different one. I'll kick myself if the lost ones turn out to be prime. "Brian J. Beesley" wrote: > On 21 Nov 99, at 23:23, Matthew Smith wrote: > > > I'm running 19.0.1. I thought that was the latest version. > > > > Matthew Smith > > Clarification to George's reply: > > Unfortunately the betas of prime95 v19.0 all reported the same > version number. (The third digit is the platform which is always 1 > for Prime95, 2 for mprime ...) > > The current version is 19.1 which you will now get by default if you > upload Prime95.zip from the "usual place". > > Regards > Brian Beesley _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Mon, 22 Nov 1999 11:51:41 -0600 From: "Willmore, David" <[EMAIL PROTECTED]> Subject: RE: Mersenne: Re: Quiet list? Is it 584, and if so, what's so intuitive? :) > -----Original Message----- > From: Steinar H. Gunderson [SMTP:[EMAIL PROTECTED]] > Sent: Saturday, November 20, 1999 6:32 AM > To: [EMAIL PROTECTED] > Subject: Mersenne: Re: Quiet list? > > On Fri, Nov 19, 1999 at 08:27:12PM -0800, Spike Jones wrote: > >The total number of posts > >I receive each day is far more predictable than the number > >of posts on GIMPS. Predictably. > > Funnily enough, when I receive nothing from GIMPS, I receive nothing > from other people either! (No, this is not a mail burp. I still receive > _some_ mail.) > > To get you something to do, here's an interesting problem. The beaty > if it, is that you could do it by brute-force, or you could simply > look at it, or see a solution and solve it in under a minute. Pen and > paper allowed only -- no calculators, computers etc. > > First, perhaps I should explain some notation :-) a_11 is the letter `a', > followed by 11 in subscript. x^2 is the letter `x', followed by the letter > 2 in superscript (ie. `x^2' would be mathematically the same as `x*x'). > OK, here goes: > > If (3x^2 - x - 2)^6 = (a_12)x^12 + (a_11)x^11 + ... + (a_1)x + a_0, what > is a_0 + a_2 + a_4 + ... + a_12? > > The answer is an integer from 0 to 999, inclusive. > > (This is just one of the problems from something called the `Abel > contest', a voluntary contest in maths open to all pupils (<21, but in > general nobody under 18 enters) in Norway. They have a tradition of > making problems requiring very little actual mathematical knowledge > (generally if you know (a+b)^2 = a^2 + 2ab + b^2 and Pythagoras, you > have 90% of what you need), but more requiring the right way of thinking. > Also, only pen and paper is used, to prevent some ways of solving the > problems... This question was number 9 (of 10), from the second round, > where the contestants (the 10% best from round one) are given 100 > minutes to try to solve the 10 questions.) > > Hope most of you see the quick solution :-) Please don't post the answer > to the list quite yet, give people their time... > > /* Steinar */ > -- > Homepage: http://members.xoom.com/sneeze/ > _________________________________________________________________ > Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm > Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Mon, 22 Nov 1999 17:30:43 -0600 From: "Willmore, David" <[EMAIL PROTECTED]> Subject: RE: Mersenne: Re: Quiet list? Oops, sorry, everyone. I didn't mean to send this to the group. Luckily, I was wrong. :) Seems I forgot how to multiply polynomials somewhere between here and high school. > -----Original Message----- > From: Willmore, David [SMTP:[EMAIL PROTECTED]] > Sent: Monday, November 22, 1999 11:52 AM > To: 'Steinar H. Gunderson' > Subject: RE: Mersenne: Re: Quiet list? > > Is it 584, and if so, what's so intuitive? :) > _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Mon, 22 Nov 1999 20:02:48 -0500 From: Matthew Smith <[EMAIL PROTECTED]> Subject: Mersenne: Problem solved. That must be it. My prime.ini file got lost. Thanks for your help. "Brian J. Beesley" wrote: > On 22 Nov 99, at 6:47, Matthew Smith wrote: > > > Okay. I just upgraded to 19.1. Prime95's first action as a newer > > entity was to unreserve yet another of my exponents. > > Odd. > > What value do you have for "days of work to get"? What is the value > of the parameter "RunningAverage" (in local.ini), and is it falling? > What type of assignments are you requesting? > > > I requested more, > > so now I have a different one. I'll kick myself if the lost ones turn > > out to be prime. > > This should balance itself out. It's about as likely that the > exponent that you got but wouldn't have requested will turn out to be > prime as the one that was returned. > > Regards > Brian Beesley _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Mon, 22 Nov 1999 17:38:53 -0800 From: "Joth Tupper" <[EMAIL PROTECTED]> Subject: Mersenne: Another polynomial Here is another polynomial question. This is from one of the American Math Competitions for high school students. I would have to do some research to validate which year and contest it is from but I think it is from the 1999 Invitational. Consider the polynomial f(x) = x^2 - 19x + 99. What is the sum of all positive integers x such that f(x) is the square of an integer? [This is also an integer between 0 and 999, inclusive.] I think the contest allows an average of 10 to 12 minutes per problem. Assume that tools are limited to pencil and paper. _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 23 Nov 1999 10:29:09 -0000 From: "Daniel Grace" <[EMAIL PROTECTED]> Subject: Mersenne: Mersennes are square free? I looked at Chris Caldwells page on Wieferich (1909) primes but I could not see exaclty how p^2|2^(p-1)-1 relates to Mersennes with square factors? I can see that Mp=3(2^(p-1)-1). So my question is this "How does one derive Wieferich's result, from the statement: let p be a prime and n be an integer such that p^2|2^n-1?" I assume that n must be a prime otherwise: Is it always true that if q|2^p-1 where p & q are primes then q^2|2^(pq)-1? eg. 23^2|2^(23.11)-1. Thanks. - ---------------------------------------------------------- Daniel e-mail: [EMAIL PROTECTED] _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 23 Nov 1999 02:55:04 -0800 From: Luke Welsh <[EMAIL PROTECTED]> Subject: Mersenne: Book Report I just finished reading "Why Computers Are Computers: The SWAC and the PC" by David Rutland Wren Publishers PO Box 1084 Philomath, OR 97370 USA ISBN 1-885391-0506 (Hardcover) 1-885391-06-4 (Softcover) Copyright 1995 The author tried to write two books at once. He really wanted to explain to the novice that the modern PC is essentially the same as the SWAC, built in 1950. By this, Rutland means that both are "stored program machines", and I skipped over sections where he tried to make this point. But I did learn that the credit does not all go to John von Neumann!! That was a huge surprise to me. Of course, I was really interested in Mersenne primes since the SWAC was the first computer to find new Mersennes, 5 of them in all were found by Robinson. Also, Alex Hurwitz cut his teeth on the SWAC, although he found M(4253) and M(4423) on an IBM 7090. The SWAC was dedicated on August 17, 1950. Regarding primes, Rutland writes on pages 126-127: "Some months after the dedication, the SWAC was running well enough for a program to be run for hours without error. The mathematicians at INA, like mathematicians throughout the world, were interested in prime numbers. Numbers like 3, 5, 7, 11, 13 and so on that cannot be divided exactly by any number but themselves. As the numbers get bigger the primes keep getting further and further apart and are more difficult to find. Each number must be tested to see if a smaller number will divide into it exactly. Mathematicians who deal with the theory of numbers are very interested in finding large ones, so they programmed the SWAC to start with a known large prime and find a larger one. It was set to this task and after hours of calculating, it came up with a prime number larger than any yet known. This was an achievement only of real interest to mathematicians. But to all of us that had built a computer from 'scratch' in less than 18 months, it was like a milestone in the history of computers." Attention Chris Caldwell! Reference Table 2 on http://www.utm.edu/research/primes/notes/by_year.html Is this a previous record? Rutland writes that a record was set in 1950. All I have seen was Miller and Wheeler setting records on EDSAC on Jun 7, 1951: http://www.scruznet.com/~luke/lit/lit_062s.htm http://www.scruznet.com/~luke/lit/lit_063.txt Aside, didn't they find 934(2^127-1)+1 and then 978(2^127-1)+1 before 180(2^127-1 )^2+1 ? Regardless, Rutland's only mention of Mersennes is on page 143: "The mathematicians continued to look for high prime numbers. The particular primes that the SWAC looked for are called Mersenne numbers, which are calculated by the formula 2^p-1. That is, 2 raised to the p power minus 1 in which p is a known prime number. For example, if p equals 3, then the Mersenne number is 2 cubed (2^3), or 8 minus 1 which equals 7, a prime. But all Mersenne numbers are not prime and must be tested to see if they can be divided by a different number than themselves and 1. The first few prime Mersenne numbers, when p is equal to 2, 3, 4 and 7, are 3, 7, 31 and 127. As p gets larger, the numbers get bigger very quickly and testing to see if they are prime also gets longer. After 453 hours, SWAC found a prime with p equal to 2281, which when written down would be over 700 decimal places long! M2281 was the 5th and last Mp that Robinson found. The figure of 453 hours is probably cumulative, and not just to test M2281 only. Robinson announced his 5 Mersenne Primes here: http://www.scruznet.com/~luke/lit/lit_024.txt the first two were found on January 30, 1952. He also wrote that it took about 5 minutes per small p and one hour for each large p. There a wealth of information in the book. Rutland was a young Electrical Engineer (Cal Tech) who helped build the SWAC. He describes it in detail and compares it to other machines of the day. Lots of great photographs. I apprectiated his description of early memories, especially Mercury delay lines and Willams tubes. Amazing! Perhaps the best of all was the shift register: just looped cables, "electrical delay lines" :-) Rutley writes they programed in hexadecimal 0 through 9 and A through F. But Robinson, http://www.scruznet.com/~luke/lit/lit_024.txt used 0 through 9 and u, v, w, x, y, z. I'm still confused about the SWAC word size. Rutley often mentions 37 bits. But then he writes "These lamps were arranged in eight groups of four for the 36 bits of the SWAC word, plus an extra lamp for the 37th bit which represented the sign of the number." Maybe each "byte" had a parity bit? I doubt it because the arithmetic unit was 37 bits wide. Are you reading this, Alex? Here's something that hasn't changed in 50 years: "When they found an error, they would of course report it to the engineers, who generally tried to blame the programmer by saying 'maybe the program is wrong.'" I'm *still* hearing that. I'm telling you, it is NOT my code! It the guard's two-way radio interfering with the USB and I can prove it!! 3 timeouts cause a reset and it's in the spec!!! Amazon.com US$24.95 3-5 weeks estimated delivery http://www.amazon.com/exec/obidos/ASIN/1885391056/qid=943260411/102-6052119-6422430 I recommend this book. - --Luke _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 23 Nov 1999 12:24:03 -0000 From: "Daniel Grace" <[EMAIL PROTECTED]> Subject: Re: Mersenne: Mersennes are square free? > I can see that Mp=3(2^(p-1)-1). I meant to say 2^(p-1)-1=(2^((p-1)/2)-1)(2^((p-1)/2)+1). Daniel. _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 23 Nov 1999 07:54:08 -0600 From: "Chris K. Caldwell" <[EMAIL PROTECTED]> Subject: Re: Mersenne: Mersennes are square free? At 10:29 AM 11/23/99 +0000, you wrote: >I looked at Chris Caldwells page on Wieferich (1909) >primes but I could not see exaclty how p^2|2^(p-1)-1 relates to >Mersennes with square factors? Did you look at http://www.utm.edu/research/primes/notes/proofs/SquareMerDiv.html ? Chris. _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 23 Nov 1999 22:04:06 EST From: [EMAIL PROTECTED] Subject: Mersenne: Mersenne Primes Paper Greetings, everybody! My real name (which I've kept more or less a secret until now) is Stephan T. Lavavej. I'm a senior in high school, and I recently had to write an Extended Essay for the International Baccalaureate program. Now that I've just turned it in, I don't need to worry about getting "help" from outside sources. So, as promised, I have made my Extended Essay available on the Internet. Its location is: http://members.aol.com/stl137/private/mersenne.zip (Please see notes below about distributing this address and my name) Its title is "Mersenne Primes: Development through History, Ongoing Work, and a New Conjecture". That mostly sums up what it discusses. Extended Essays are required to be approximately 4000 words long; this is NOT a simple report for math class. I hope that there is something of interest in the paper for everyone, be it the somewhat detailed history of Mersenne primes, the description of the GIMPS work, or the discussion of my conjecture. My conjecture has been discussed on this list, but the paper includes graphs and detailed explanations of what I did. If anything, read it for the predictions of the 40th, 50th, and 60th Mersenne primes! Some miscellaneous things: The file is in .ZIP format. Once unzipped, Microsoft Word 97 will view it (I use no weird fonts, only Times New Roman). I have virus-checked the .DOC file with Norton Antivirus 2.0 (with the latest virus definitions update), but those more paranoid than me can scan it too. The URL provided is in my private directory; as such, please do not post it on publicly available websites. Feel free to send the URL to interested acquaintances. Likewise, please do not link my AOL screen name with my real name Stephan Lavavej publicly, but feel free to use it on this list and in personal E-mails. Of course, *grin* I would like nothing better than to see my conjecture in _Unsolved Problems in Number Theory, 3rd Edition_ by Richard K. Guy. (I can dream, can't I?) Mentions of my conjecture in (for example) the Mersenne FAQ are okay as long as they doesn't mention my screen name ("Stephan Lavavej conjectures that..." is absolutely okay.) I figured that a determined enough wacko could find out my real name anyways, you see. I should at least repeat my acknowledgements on this list: "Thanks to G. Woltman and S. Kurowski for sparking this journey, to Wolfram Research and Texas Instruments for producing their excellent mathematical tools, and special thanks to A. Lee for help in debugging this paper." Also, without the invaluable assistance of L. Wiman, I would not have understood the difference between approximation double-squigglys and asymptotic single-squigglys "~". Without the Mersenne mailing list, it would have taken me much longer to write the section about my conjecture. (I based that section on an E-mail I sent the list. Why do work twice?) :-D I hope you enjoy reading my paper! Stephan "Now you know what those initials stand for" Lavavej _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 24 Nov 1999 10:29:27 +0100 (MET) From: [EMAIL PROTECTED] Subject: Re: Mersenne: Mersennes are square free? Daniel Grace <[EMAIL PROTECTED]> asks > I looked at Chris Caldwells page on Wieferich (1909) > primes but I could not see exaclty how p^2|2^(p-1)-1 relates to > Mersennes with square factors? I can see that Mp=3(2^(p-1)-1). > So my question is this "How does one derive Wieferich's result, > from the statement: let p be a prime and n be an integer such > that p^2|2^n-1?" > I assume that n must be a prime otherwise: > Is it always true that if q|2^p-1 where p & q are primes > then q^2|2^(pq)-1? eg. 23^2|2^(23.11)-1. Write 2^p = 1 + k*q, since you assume q | (2^p - 1). You want to show that q^2 divides (1 + k*q)^q - 1. Use the binomial theorem. Or show by induction on m that (1 + k*q)^m == 1 + k*q*m (mod q^2) for all integers m >= 0. Then set m = q. _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ End of Mersenne Digest V1 #663 ******************************
