Mersenne Digest Wednesday, February 9 2000 Volume 01 : Number 690 ---------------------------------------------------------------------- Date: Tue, 8 Feb 2000 16:18:30 -0000 From: "Brian J. Beesley" <[EMAIL PROTECTED]> Subject: Re: Mersenne: Hypothesis On 7 Feb 00, at 22:40, Wojciech Florek wrote: > > Hi all! > Due to some reasons I've considered numbers in a form > 3*2^n (3,6,12,....) > and I've found that almost in each interval 3*2^n..3*2^(n+1) > there are one, two or three exponents of Mersenne prime. Isn't this really just saying that Mersenne primes have a similar distribution to the series k*2^n? I thought we already had a hypothesis suggesting that there should be about 1.4 Mersenne primes per octave - on average - which is actually a slightly more informative version of the hypothesis based on the observation reported here. > The first two: 2,3 are below or equal 3*2^0. > `Almost' means that there is a true gap for n=6: > there are no exponents between 3*64=192 and 3*128=384. So the hypothesis has a counter-example ... Sorry, I'll try to be more constructive. If you have a hypothesis that there are, on average, k Mersenne primes per octave but that the distribution is random, if you sample the number of Mersenne primes per octave (starting at _any_ point) then you should get something like a Poisson distribution with mean k. It might be, from the limited sample we have, starting the sampling interval at 3*2^n (as opposed to q*2^n for some other q) gives a better/smoother fit than others - I don't know, I haven't tried - but, in any case, the sample size we have to go on is pitifully small for testing the hypothesis. > The other possible gap is for n=20 3*2^20=3145728..6291456, > but this is a reminder of the v17 bug (???). We haven't fully searched this interval yet, and double-checking is nowhere near complete. I'd guess that the bad v17 results have been redone long since. > Among 36 considered exponents (without 2,3) 25 can be written > as 3*2^n+p OR (sometmes AND) 3*2^n-p, where p is a prime. > On the other hand, > 11 exponents are expressed as 3*2^n +/- c, where c is a composite > number. I've considered only differences with interval > limits. The smallest is 2203=3*512+(5*149)=3*1024-(11*79). > The others are: 2281,11213,44497,86243,110503,132049,216091, > 756839,859433,1257787,2976221. Note that the two largest known > exponents are > 3021377=3*2^20-124351 [prime!] 6972593=3*2^21+681137 [prime!] This is an interesting observation. Do we have a handle on how likely it is that an arbitary number of a similar size can be represented this way? I have a (probably incorrect, gut) feeling that the "composites" are under-represented. However, even if we found a relationship like this which is true for all known Mersenne primes, we wouldn't be sensible to use it as a criterion for eliminating exponents without a decent proof (not just a hand-waving argument) as to why the relationship _must_ hold for Mersenne prime exponents. Regards Brian Beesley _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 8 Feb 2000 17:57:03 +0100 From: "Hoogendoorn, Sander" <[EMAIL PROTECTED]> Subject: RE: Mersenne: Request for feature As long as we're asking, is it possible to slit up the Assignments Report and the Cleared Exponents Report into seperate lists for double checking,first time checking and factoring, these lists get to large to download often. - -----Original Message----- From: Alan Vidmar [mailto:[EMAIL PROTECTED]] Sent: maandag 7 februari 2000 18:26 To: [EMAIL PROTECTED] Subject: Mersenne: Request for feature Scott, Would it be possible to add a CPU type/speed column to the "Exponents Assigned" list of the "Individual Account Report"? This info seems to be collected as the "Machines Assigned to PrimeNet" suggests. Thanks, Alan _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 8 Feb 2000 11:50:45 -0500 From: "Conor McCutcheon" <[EMAIL PROTECTED]> Subject: Re: Mersenne: AMD Athlon problems Well, I wish it was as simple as power management, but turning that off in the bios and windows is the very first thing I did after installing the CPU. As for the CPU overheating, I really doubt it, but it is worth looking into. I have what appears to be a good fan attached to the CPU, one designed to be used the way I am using, with the heat transfer goop on it as well, and I have an additional case fan that brings in cool air from the room. All appear to be working. As for the suggestion that the CPU speed is incorrectly set, it is not. I have checked that setting and played with it enough to be sure that it is correct. It is also worth noting that the number of cycles each iteration takes matches extremely well with the time reading if and only if my CPU is set to 750 in the preferences. At this point I think it might be something with my memory, because that could explain the CPU running at full speed but not mprime or prime95. Does this seem like a reasonable explanation (I have 128M of PC100 sdram in one dimm)? The Athlon 550 I mentioned only had PC133 (or was it 166), but I can't imagine that making a 550 outperform a 750 by nearly 10 times. Anyway, thank you again for your help. - -Conor - ----- Original Message ----- From: "Brian J. Beesley" <[EMAIL PROTECTED]> To: "Conor McCutcheon" <[EMAIL PROTECTED]> Sent: Tuesday, February 08, 2000 11:18 AM Subject: Re: Mersenne: AMD Athlon problems > On 7 Feb 00, at 23:25, Conor McCutcheon wrote: > > > I have a new athlon 750 running either prime95 or mprime 24 hours a day, > > but I am getting a ridiculously slow per iteration time for a 9.5 million > > range exponent (1.192 sec avg). > > Try turning off the power management, it's possible that the system > is running in "slow" mode due to apparent non-activity of keyboard > etc. > > The M/B should have temp sensors for the CPU. If you can't read these > whilst Prime95/mprime is running (usually the M/B supplier does > include a disk with a Windoze monitor program) you should be able to > read the CPU temp from the BIOS. If the CPU gets too warm then it > will probably drop its speed. > > I believe Athlons do require a special cooling fan & PSU, if the > system was supplied with substandard components then it could quite > easily be overheating. > > > Regards > Brian Beesley _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 08 Feb 2000 12:29:20 -0500 From: George Woltman <[EMAIL PROTECTED]> Subject: Re: Mersenne: Strange result line inprime V19.2 Hi, At 05:11 PM 2/8/00 +0100, Grieken, Paul van wrote: >Some days ago prime V19.2 finished an exponent, the line looks like > >UID: grieken/C8B74DDA1, M7563337 is not prime. Res64: CBE45D3443C7D394. WV1: >9BE40783,5609919,80000000 > >I have two questions: >1. what is the number behind my UID, never seen in other lines The C8B74DDA1 is a randomly generated computer ID. Very handy for users with a large number of machines. >2. What means the last number, the one that begins with an 8. >thanks for any reaction The 8 means that this exponent was started with a previous version of prime95. The 7 zeroes indicate that no errors were detected during the LL test. Regards, George _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 8 Feb 2000 12:54:06 -0500 From: "Ethan O'Connor" <[EMAIL PROTECTED]> Subject: RE: Mersenne: AMD Athlon problems - -----Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]]On Behalf Of Conor McCutcheon >I have a new athlon 750 running either prime95 or mprime 24 hours a day, but >I am getting a ridiculously slow per iteration time for a 9.5 million range >exponent (1.192 sec avg). I have installed on prime95 on over 20 machines now, >including an Athlon 550 (it gets about .2 something seconds on a 9.4 million range >exponent), and experience tells me this is ridiculous. Have you tried running one of the programs which verify that your L2 cache is actually functioning (despite what the settings in the BIOS may claim)? http://www.softseek.com/Utilities/Benchmarking_and_Tune_Up/Review_22226_inde x.html is one. I remember hearing that the Athlon 750 had issues with L2 cache at least for a while. I think they run it at ~40% of the core clock rather than 1/2 as with the slower Athlons. That alone obviously doesn't explain your results but maybe if the cache is not functioning at all for some reason... ? - -Ethan O'Connor [EMAIL PROTECTED] _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 09 Feb 2000 09:14:56 +1300 (NZDT) From: Bill Rea <[EMAIL PROTECTED]> Subject: Mersenne: 60 Day Expiration mikus wrote:- >I've mothballed a middling-speed non-Intel machine. That machine >could have been participating in GIMPS, but I chose not to have >it do so any more. The reason - I resent feeling "pressured" by >expiration requirements and contact-every-xx-days requirements. I'm a bit mystified by this comment. I use a number of SPARCs for GIMPS work. I just check out an exponent and return the result when it is finished using the manual assignment pages. The last exponent I checked out for a slow machine was 9519571. I expect that will take about 70 days to do the LL test. I certainly don't feel any pressure. I'm not aware of any requirement to contact the server at any particular interval. I've just checked my assignment status and the above exponent will not expire for another 112.2 days. My main problem is that one other system person has decided to start running setiathome on several of the systems I've been using for GIMPS. This really kills interation times. "Friendly" discussion hasn't resulted in any solution yet. _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 08 Feb 2000 15:37:41 -0500 From: Jeff Woods <[EMAIL PROTECTED]> Subject: Mersenne: Synchronization -- some oldies in the list Why, when there's a synchronization, do some VERY old results get left in my "cleared since last sync" report? - ------- Exponents Cleared since last Synchronization ------- prime fact Lucas-Lehmer residue or factor exponent bits [residues partially masked] date returned computer ID - -------- ---- -- -------------------------------- --------------- - ------------ 4271591 61 D 0x81E4CE076ABF61__ 03-Feb-00 18:18 PALPATINE 4271611 61 D 0x6DBD111507DD6A__ 05-Feb-00 18:27 WOOKIEE 4298507 61 D 0x00276C26EE0C80__ 05-Feb-00 18:53 JEDIMASTER 5568359 63 0x918895A0B8E691__ 14-Aug-99 09:30 YODA 5665069 63 0x166178B2710599__ 08-Sep-99 03:25 SKYWALKER 5761841 63 0x6C0FD089F3899C__ 13-Aug-99 15:01 SKYWALKER 5854777 63 0x0A011A3B186AB5__ 27-Aug-99 17:00 YODA 6623549 63 0x3ABF39E11D0E97__ 12-Sep-99 04:29 ENDOR 6658511 63 0x5ADA9831B6BC88__ 23-Sep-99 10:58 EMPEROR 8742157 63 0x62DF3CA29A7A4A__ 07-Feb-00 15:32 ENDOR Why all the August and September results still in the list, if a synchronization just occurred? _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 8 Feb 2000 15:45:32 EST From: [EMAIL PROTECTED] Subject: Mersenne: Re: out to pasture Mikus Grinbergs writes: >As long as processing one exponent took less than 60 days, I was >willing to chance it that my exponent would not be reassigned >before that machine finished it. But now processing of larger >exponents takes more than 60 days. I am __NOT__ willing to >double my effort (i.e., make contact when half through) just to >"keep" my exponent from being reassigned. My solution - stop >participating with that machine. Since this is a non-x86 machine you're talking about, you'd be using manual test mode. If you'd looked at the GIMPS manual test checkout form lately, you'd have seen that Scott has increased the maximum time for an assignment to 120 days, precisely because first-time LL tests are now taking so long. I've run several exponents in the 7-8M range on my trusty old 200MHz Alpha 21064, but now have it doing double-checking since it gets better relative performance at those smaller FFT lengths. (But I'm working on Mlucas 3.0, which will hopefully provide a nice boost, thus breathing new life into some of these older CPUs). An exponent in the 4.2M range takes just two weeks, which is perhaps not fast enough to satisfy the thrills-a-minute crowd, but fast enough for me to be satisfied that even this oldest of my machines is contributing. - -Ernst ftp://209.133.33.182/pub/mayer/README.html _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 8 Feb 2000 15:45:38 EST From: [EMAIL PROTECTED] Subject: Mersenne: Re: details...must...have...details... Alan Vidmar wrote: >Would it be possible to add a CPU type/speed column to the >"Exponents Assigned" list of the "Individual Account Report"? I second that, and move that a program/version line also be added, to both the individual account report and the cleared exponents list. But Scott is preparing to move his compute stuff and his household and to take over the reigns of the fully-fledged and capitalized entropia.com (and I wish him well! Remember us when you're an Internet Billionaire, Scott...), so these types of things are likely not high on his present priority list. - -Ernst _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 8 Feb 2000 19:46:20 EST From: [EMAIL PROTECTED] Subject: Re: Mersenne: The return of poaching? > Yes, so the *real* milestone is less than 10% away. > The real milestone is not 10^12 but rather 2^40 ops > per second. Lets get with it! My model predicts > we will hit 2^40 in May. {8-] spike What's this number "40"? Shouldn't it be 2^(2^6-1) ? :) _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 8 Feb 2000 22:09:36 -0500 From: "Conor McCutcheon" <[EMAIL PROTECTED]> Subject: Mersenne: AMD Athlon problem solved, no thanks to AMI bios Thanks to everyone for their help, the problem is resolved. The basic problem was an utter lack of L2 cache. It made sense that L2 cache was the problem but I could not figure out how to turn it on, because I had experimented with nearly every concievable setting in the BIOS that even smelled like it related to no avail. Finally, after looking for a good while for more specific diagnostic tools I finally just decided to update my bios despite the fact that nothing in the change-log mentioned the caches. Anyway, the external cache option suddenly changed from write-thru to write-back or vice-versa (I don't remember now) and that solved the problem. You would think FIC (MB manufacturer) would mention a defunct cache as a bios bug. Thanks again, I just thought I should mention it in case someone else is affected by this problem. - -Conor McCutcheon _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 08 Feb 2000 20:12:34 -0800 From: Spike Jones <[EMAIL PROTECTED]> Subject: Re: Mersenne: The return of poaching? [EMAIL PROTECTED] wrote: > > per second. Lets get with it! My model predicts > > we will hit 2^40 in May. {8-] spike > > What's this number "40"? Shouldn't it be 2^(2^6-1) ? :) ummm, no. but i would buy 2^(2^(2^2+1)+2^(2^1+2^0)) {8^D spike _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 09 Feb 2000 00:06:10 -0600 From: [EMAIL PROTECTED] Subject: Re: Mersenne: pi Hi, I have been considering the possible role pi might play in the progression of mersennes. It is generally accepted that the value of pi is a never ending series. But when I look at the circle, the formula for the area of a circle with a radius of 6 inches is: A=pi*r^2 = 3.1416 * (6)^2 = 113.0976. We did not, however, use the full and correct expansion of pi in the calculation. Pi has been figured out to over a billion (not sure of the exact figure) digits with no apparent end or pattern. But when I look at a circle I see a finite area within the circle with no means of growing or escape. Logic seems to indicate that pi would have to be a finite exact value since the area in the circle is finite. So, either the figure for pi is in error (not likely) or pi has a end. The end. What say ye? Dan _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 9 Feb 2000 02:21:27 -0500 From: "Vincent J. Mooney Jr." <[EMAIL PROTECTED]> Subject: Re: Mersenne: pi Quoting from Dan: "Logic seems to indicate that pi would have to be a finite exact value since the area in the circle is finite. So, either the figure for pi is in error (not likely) or pi has a end." No, this might be called one of the pathologies of mathematics. What seems to be so isn't. It is certain that Pi is a "never ending series" as you put it. Perhaps this will help. Sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... to infinity as the fractions become smaller and smaller (1/(2^n) as n increases without limit). The sum is 2. Can you Dan accept that a never ending sum of smaller and smaller terms has a precise finite value? It is an integer at that. Plus we never get to add all the terms -- there is always just one more and it would take infinite time to add the infinity of terms. Now look at 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... to infinity as the fractions become smaller and smaller. The sum is infinity, that is, it never stops increasing. Can you Dan accept that this sum of smaller and smaller terms has no precise value as it slowly and endlessly grows larger (hence infinity)? And there is a neat or "pathological" property of this infinitely large sum. Let's say we get larger than integer N after a million billion billion terms. So now we are adding 1/(million billion billion + 1), then 1/(million billion billion + 2), then 1/(million billion billion + 3), then + 4 etc. As the sum crawls toward N + 1, we are less than .000 000 000 000 000 000 000 001 away from N + 1 eventually (an American billion being 9 zeros). The sum never "lands on" exactly N + 1 and skips landing on all integers (and there are an infinity of those). In fact, to get as close as one wants, say .000 (million more zeros) 01, to an integer, another N' is needed where N' is >> N (N' is much larger than N). The partial sum can be made as close to an integer as we like. But the partial sum is never exactly an integer. In other words, for all integer M, the fixed sum 1 + 1/2 + 1/3 + 1/4 + .... + 1/M (a fixed sum because M is the last one in a finite summation) is never an integer even though the infinite series (as M grows to infinity) passes through all integers. The better mathematicians in this group (that is, all other mathematicians :-) may give a better explanation. At 12:06 AM 2/9/00 -0600, Dan wrote: >Hi, I have been considering the possible role pi might play in the >progression of mersennes. It is generally accepted that the value of pi is >a never ending series. > >But when I look at the circle, the formula for the area of a circle with a >radius of 6 inches is: A=pi*r^2 = 3.1416 * (6)^2 = 113.0976. > >We did not, however, use the full and correct expansion of pi in the >calculation. > >Pi has been figured out to over a billion (not sure of the exact figure) >digits with no apparent end or pattern. > >But when I look at a circle I see a finite area within the circle with no >means of growing or escape. Logic seems to indicate that pi would have to >be a finite exact value since the area in the circle is finite. > >So, either the figure for pi is in error (not likely) or pi has a end. > >The end. >What say ye? >Dan > _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 09 Feb 2000 02:31:09 -0500 From: gav <[EMAIL PROTECTED]> Subject: Re: Mersenne: pi I think my favorite counterexample to arguments like this is Gabriel's Horn. Take the function 1/x, and revolve it around the x-axis. You now have something that looks very similar to a trumpet's bell. Now, find the volume of this from 0 to infinity. It has a finite volume. However, it has an infinite surface area. (These can both be determined by integration, however, it's been long enough since my last Calc class that I'd probably mess up the integral for surface area, so I won't try...) Also, spend a bit of time with the concept of limits. Limits can approach, to an infinitesimally small difference, a finite value. So think of the circle calculation as a limit problem, something along the lines of: lim(y->pi)y * r^2 = area of circle. Since we know the area of the circle to be finite, we know the limit must be finite. However, this makes no stipulations on the properties of y (and essentially pi, from that standpoint), so in no way is pi limited to having an infinite number of decimals. And then there's just all the standard research on pi. Check out the arctan function, and some of the newer "distributed calculating digits of pi" projects on the web for other formulas that can solve for any digit (or up to and including that digit) of pi. Especially looking at the summations, it's somewhat apparent that they don't resolve as simply as all that. My 1 3/4 cents. George At 12:06 AM 2/9/00 -0600, [EMAIL PROTECTED] wrote: >Hi, I have been considering the possible role pi might play in the >progression of mersennes. It is generally accepted that the value of pi is >a never ending series. > >But when I look at the circle, the formula for the area of a circle with a >radius of 6 inches is: A=pi*r^2 = 3.1416 * (6)^2 = 113.0976. > >We did not, however, use the full and correct expansion of pi in the >calculation. > >Pi has been figured out to over a billion (not sure of the exact figure) >digits with no apparent end or pattern. > >But when I look at a circle I see a finite area within the circle with no >means of growing or escape. Logic seems to indicate that pi would have to >be a finite exact value since the area in the circle is finite. > >So, either the figure for pi is in error (not likely) or pi has a end. > >The end. >What say ye? >Dan > > > >_________________________________________________________________ >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: Tue, 8 Feb 2000 23:40:57 -0800 From: "Scott Kurowski" <[EMAIL PROTECTED]> Subject: Mersenne: PrimeNet Top Producers List Hi all, Following the database synchro we performed on PrimeNet yesterday, we cleaned up some of the 'dead' user accounts over a year old. The cumulative machine times were added to the Entropia.com, Inc. 'challenge' account, the first one opened on PrimeNet in April 1997. Hopefully nobody will mind our reclaiming the fragmented time. :-) regards, scott [EMAIL PROTECTED] _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 9 Feb 2000 09:54:50 +0100 (CET) From: Henrik Olsen <[EMAIL PROTECTED]> Subject: Re: Mersenne: pi On Wed, 9 Feb 2000 [EMAIL PROTECTED] wrote: > Hi, I have been considering the possible role pi might play in the > progression of mersennes. It is generally accepted that the value of pi is > a never ending series. > > But when I look at the circle, the formula for the area of a circle with a > radius of 6 inches is: A=pi*r^2 = 3.1416 * (6)^2 = 113.0976. > > We did not, however, use the full and correct expansion of pi in the > calculation. > > Pi has been figured out to over a billion (not sure of the exact figure) > digits with no apparent end or pattern. > > But when I look at a circle I see a finite area within the circle with no > means of growing or escape. Logic seems to indicate that pi would have to > be a finite exact value since the area in the circle is finite. Yep you're actually right, pi has a finite exact value. The problem isn't that Pi isn't finite, it's less than 4 so it's finite. The problem isn't that it isn't exact. The problem is that it can't be represented exactly in decimals which mens that when we write the expansion, we'll always have to make do with an approximation to the exact value. > So, either the figure for pi is in error (not likely) or pi has a end. Any decimal representation of pi is in error, since it can only be an approximation. > The end. > What say ye? > Dan I think where your argument slips is in confusing the number for it's representation, ie. how it's written. These are two different concepts, and confusing them leads to argumenting from false analogies. - -- Henrik Olsen, Dawn Solutions I/S URL=http://www.iaeste.dk/~henrik/ Linux isn't at war. War involves large numbers of people making losing decisions that harm each other in a vain attempt to lose last. Linux is about winning. Alan Cox on linux-kernel _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 09 Feb 2000 22:42:32 +1300 From: "Halliday, Ian" <[EMAIL PROTECTED]> Subject: Mersenne: Finite, Amicable, Pi...lots of topics In Unsolved Problems in Number Theory, Richard K Guy says of Mersenne primes: "their number is undoubtedly infinite, but proof is hopelessly beyond reach". He then offers some suggestions for the size of M(x), the number of primes p <= x for which 2^p -1 is prime. Gillies suggested M(x) ~ c ln x Pomerance suggested M(x) ~ c ( ln ln x ) ^ 2 This is very serious indeed, especially for those of us who believe the number of Mersenne primes to be finite. It's a fairly old book in a manner of speaking: in 1981 he poses the question as to whether 2^p - 1 is always square-free. I'm sure this has been discussed here from time to time - did we ever get an answer? In this case, Guy believes that the answer is no, and that it could be settled by computer if you were lucky. I'm still on topic if I talk about perfect numbers, where the sum of the factors of a perfect number n, which I call s(n) is equal to n. However I'm off topic as soon as I start talking about amicable numbers, sometimes called semi-perfect numbers. For a pair of amicable numbers m and n we have s(m) = n and s(n) = m. For example s(220) = 284 while s(284) = 220. I choose to mention these because of the recent mention of hairy and smooth numbers and in the context of Esau and Jacob, also recent players here, as the number 220 is of some significance in their story in Genesis 32:14. The recent heroes in this field are H J J te Riele, who "knows everything about amicable numbers" according to a now forgotten usenet poster and Lee and Madachy, who published "The history and discovery of Amicable Numbers" in the Journal of Recreational Mathematics in 1972, along with an alarmingly long list of then known amicable numbers. (Does anybody know if this journal is still published? When I subscribed to it for a while, though, it wasn't too recreational, and seemed obsessed with repunits for a while.) There are far more amicable pairs known than even perfect numbers, yet Guy's claim on their infinite number or otherwise is, surprisingly, weaker. "It is not known if there are infinitely many, but it is believed that there are." Finally, pi. Along with others, I have been amused by the reputed Alabama legislature decision, and spend a lot of time looking at the urban legends at http://www.snopes.com/ which is one of the most significant sites on the web, possibly second only to http://www.mersenne.org/prime.htm ? However, as I believe in the inerrancy of scripture, I obviously have a problem with 1 Kings 7:23. I don't believe either that pi = 3 or that God thinks pi = 3. So, what happens? At http://www.khouse.org/articles/biblestudy/19980401-158.html we can learn that there is a subtle difference in the text from what might be expected in that the word for circumference "qav" has been replaced by the word "qaveh". If we take note of the numerical values associated with these words, qav = 100 + 6, while qaveh = 100 + 6 + 5. Accordingly, we take the implied multiplicand of 3 and extend it by 111/106, which gives an approximation of 333/106, which is 3.141509... which is accurate enough for practical purposes. Possibly not for rocket science, but that's not what we're talking about here. K House probably don't phrase their explanation in the way I would choose, but it nevertheless makes compelling reading from a reasonably mainstream source. Over history, there have been numerous other approximations to the value of pi. Our current culture seems to favour 22/7 as an approximation, and the Biblical approximation above suggests 333/106. However, this is not the best available in three digits, which is, so far as I know, 355/113, which is correct to an astonishing one part in ten million. I understand that in certain quarters, 3 1/7 was not in vogue, with 3 1/8 favoured. What, argued these particular mystics, could be a better number than five squared shared by two cubed? N P Smith asked whether we should be more concerned by those who serious propose answers which are clearly wrong or by those who spend time in repeatedly refuting these spurious claims. As for squaring the circle, another popular pastime, the Greeks noted that a square of side 8 have pretty much the same area. This points to 256/81 or sixteen squared shared by nine squared if you like that sort of thing. It's still not exact. That's what irrational means... I'm sorry to have strayed off topic: at the moment I can't find any legitimate connection between pi and Mersenne numbers - if anybody can do so then obviously I am absolved because this posting will have been on topic after all. I am absolved! Between researching this article and posting it, others have started to explore the possibility of such links. Regards, Ian W Halliday Wellington, New Zealand - --- Happiness is just around the corner. - D H Lehmer _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 09 Feb 2000 09:28:20 -0500 From: Jud McCranie <[EMAIL PROTECTED]> Subject: Re: Mersenne: Finite, Amicable, Pi...lots of topics At 10:42 PM 2/9/00 +1300, Halliday, Ian wrote: >This is very serious indeed, especially for those of us who believe the >number of Mersenne primes to be finite. Why would any of us believe that there are only a finite number of Mersenne primes? >It's a fairly old book in a manner of speaking: in 1981 There's a newer edition, 1994. +--------------------------------------------------------+ | Jud McCranie | | | | 137*2^197783+1 is prime! (59,541 digits, 11/11/99) | | 137*2^224879+1 is prime! (67,687 digits, 1/00) | +--------------------------------------------------------+ _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 9 Feb 2000 07:40:37 -0700 From: "Aaron Blosser" <[EMAIL PROTECTED]> Subject: RE: Mersenne: pi > The problem isn't that Pi isn't finite, it's less than 4 so it's finite. > The problem isn't that it isn't exact. > The problem is that it can't be represented exactly in decimals which mens > that when we write the expansion, we'll always have to make do with an > approximation to the exact value. Consider this: Let's assume that the universe is spherical (a logical assumption if we assume it's the result of a currently expanding explosion xx years ago). If we were to calculate the radius of this sphere down to a single atomic width, using some decently expanded version of pi would could come up with an exact number for the volume of the universe. What I'm getting at is that at some point, pi reaches a practical limit at which expanding more decimal points is an abstraction because we could never measure anything large enough for it to be useful. I mean, c'mon! The universe is only so big! :-) Being in a hurry, I don't have the time to figure out how many decimal places that would be...perhaps someone more adventurous would care to give it a go. Aaron _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 09 Feb 2000 09:22:15 -0500 From: Jud McCranie <[EMAIL PROTECTED]> Subject: Re: Mersenne: pi At 12:06 AM 2/9/00 -0600, [EMAIL PROTECTED] wrote: >But when I look at a circle I see a finite area within the circle with no >means of growing or escape. Logic seems to indicate that pi would have to >be a finite exact value since the area in the circle is finite. No, pi is irrational, which means that the digits go on forever without repeating. >So, either the figure for pi is in error (not likely) or pi has a end. The calculated value of pi is never exact, since it is calculated to a finite precision. +--------------------------------------------------------+ | Jud McCranie | | | | 137*2^197783+1 is prime! (59,541 digits, 11/11/99) | | 137*2^224879+1 is prime! (67,687 digits, 1/00) | +--------------------------------------------------------+ _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 09 Feb 2000 10:50:44 -0500 From: Jeff Woods <[EMAIL PROTECTED]> Subject: Re: Mersenne: pi You're bumping up against the Fundamental Theorem of Calculus here. Pi DOES have a precisely defined value, but you cannot express it in decimal form. You can express it as an infinite expansion, however. Just as you can never get to the end of pi, though its value is known, you can never PRECISELY note the area of a circle -- you can only express it more and more accurately, depending on how accurate the value of PI you use is. Thus, the limit of the area of a circle as your approximation for pi approaches an infinite expansion is pi*r^2. At 12:06 AM 2/9/00 -0600, you wrote: >Hi, I have been considering the possible role pi might play in the >progression of mersennes. It is generally accepted that the value of pi is >a never ending series. > >But when I look at the circle, the formula for the area of a circle with a >radius of 6 inches is: A=pi*r^2 = 3.1416 * (6)^2 = 113.0976. > >We did not, however, use the full and correct expansion of pi in the >calculation. > >Pi has been figured out to over a billion (not sure of the exact figure) >digits with no apparent end or pattern. > >But when I look at a circle I see a finite area within the circle with no >means of growing or escape. Logic seems to indicate that pi would have to >be a finite exact value since the area in the circle is finite. > >So, either the figure for pi is in error (not likely) or pi has a end. > >The end. >What say ye? >Dan > > > >_________________________________________________________________ >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: Wed, 9 Feb 2000 16:49:09 +0100 From: "Grieken, Paul van" <[EMAIL PROTECTED]> Subject: RE: Mersenne: pi I am not a math man but I follow this discussion. Can we say the same to the case of 10 divide by 3 , The result is 3 1/3 but as a decimal way of writing you never end. Maybe I just miss the whole thing, in that case I am sorry, and will continue just reading this kind of topics. Bye, Paul van Grieken > -----Original Message----- > From: Jud McCranie [SMTP:[EMAIL PROTECTED]] > Sent: Wednesday, February 09, 2000 3:22 PM > To: [EMAIL PROTECTED] > Cc: [EMAIL PROTECTED]; [EMAIL PROTECTED] > Subject: Re: Mersenne: pi > > At 12:06 AM 2/9/00 -0600, [EMAIL PROTECTED] wrote: > > >But when I look at a circle I see a finite area within the circle with > no > >means of growing or escape. Logic seems to indicate that pi would have > to > >be a finite exact value since the area in the circle is finite. > > No, pi is irrational, which means that the digits go on forever without > repeating. > > > >So, either the figure for pi is in error (not likely) or pi has a end. > > The calculated value of pi is never exact, since it is calculated to a > finite precision. > > > +--------------------------------------------------------+ > | Jud McCranie | > | | > | 137*2^197783+1 is prime! (59,541 digits, 11/11/99) | > | 137*2^224879+1 is prime! (67,687 digits, 1/00) | > +--------------------------------------------------------+ > > _________________________________________________________________ > 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 ------------------------------ End of Mersenne Digest V1 #690 ******************************
