Mersenne Digest Wednesday, October 13 1999 Volume 01 : Number 642 ---------------------------------------------------------------------- Date: Tue, 12 Oct 1999 22:53:18 -0400 (EDT) From: Darxus <[EMAIL PROTECTED]> Subject: probability of primeness (was: Re: Mersenne: splitting up 10m digit primes) I'm hoping what I have to say in this email might be important. On Tue, 12 Oct 1999, George Woltman wrote: > At 04:12 PM 10/12/99 -0400, you wrote: > >> >And how is the probability of finding a prime calculated ? > >> > >> It is roughly how-far-factored-in-bits * 2 / exponent > > > >Okay.. what's "how-far-factored-in-bits" mean ? > > I think trial factoring is done to 2^68 for an exponent around 33 million. > Thus your chance is 2 * 68 / 33000000. Okay, so as far as we know, each number is equally likely to be prime, and this probability is just based on how much has already been tested ? I'd been meaning to chart p for all mersenne primes (n^p)-1. Ugh... I apparently had bad math teachers, and GIMPS is really making me feel it. I *really* wanna play with these numbers, but I feel intellectually cripled. I charted p.. with the value of p on the y axis, and the number of the prime on the x axis. I omitted the latest one (the 2million-some digit one), since we don't know what number it actually is, since everything smaller than it hasn't been tested. I took the numbers from http://www.utm.edu/research/primes/mersenne.shtml#known The chart ooked interesting. Approximately exponential. But it didn't fit real cleanly. So I stared at it for a long time. I noticed that the last 6 mersenne primes appeared to be in pairs... #32 756839 #33 859433 #34 1257787 #35 1398269 #36 2976221 #37 3021377 Each pair seemed too close to each other for it to be random chance. So I split up the numbers... I put the odd p's (by number order in which they were discovered) in 1 column, and the even p's in another column, and charted them. I want you to do this right now. Please. In Excel, or something (what else might you use?). At least with this last 6. Like: 756839 859433 1257787 1398269 2976221 3021377 And graph each column as a seperate line on the same graph. They match up. They're like, on top of each other. If you do this for all known mersenne primes, the lines are off by one... remove the 1st even p, or add 1 to the odd p's to get the lines to be practically on top of each other (is this an argument that 1 should be on the list?). Does anybody see what I'm talking about ? Is there any significance to this ? Has somebody already written extensive papers on this ? I don't have access to a charting program this instant, or I'm sure I'd be up the rest of the night playing with this. Hopefully I'll get some time to do so tomorrow. Ugh, I can't wait. I really wanna graph the difference between the pairs... well, I have, but not with them in 2 sets... I'm hoping there's a relationship between the pairs, and a relationship between the distances between the pairs. __________________________________________________________________ PGP fingerprint = 03 5B 9B A0 16 33 91 2F A5 77 BC EE 43 71 98 D4 [EMAIL PROTECTED] / http://www.op.net/~darxus Join the Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 22:58:14 -0400 (EDT) From: Darxus <[EMAIL PROTECTED]> Subject: RE: Mersenne: splitting up 10m digit primes On Tue, 12 Oct 1999, Rick Pali wrote: > From: [EMAIL PROTECTED] > > > How about an option when you hit "QUIT GIMPS" to > > upload your P and Q files to Primenet, so someone > > can at least finish the job? > > I'm running an exponent in the 33 million area and the save-files are over > seven megabytes in size! That would require no small amount of server > space... Wow. I'd been assuming this was basically being done. How unfortunate. I believe I would not have stopped processing the number I was working on to start a 10m digit number if I'd known this... I would have just set it to work on a 10m digit number as soon as it finished the current one. It might be good to make this very clear.. maybe in the client or something, when somebody opts to give up on a number, that their work so far on it will be lost. __________________________________________________________________ PGP fingerprint = 03 5B 9B A0 16 33 91 2F A5 77 BC EE 43 71 98 D4 [EMAIL PROTECTED] / http://www.op.net/~darxus Join the Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 19:58:47 -0700 From: Eric Hahn <[EMAIL PROTECTED]> Subject: Re: Mersenne: Factoring numbers... Jukka Santala wrote: >Is it just me, or does factoring smaller Mersenne numbers take >propotionally much longer? I would expect M727 to be much faster >than the 33M range to a fixed depth, yet the opposite _seems_ to >be true. For any given factoring bit-depth, larger exponents will take a shorter period than smaller exponents. This is due to the number of potential factors that are available in at any given depth. Each bit-depth takes twice the amount of time to test as the previous one (ie: 2^57 takes 2x the length of 2^56) To determine the approximate number of potential factors that must be tested at any given depth, take the bit-size of the first pontentail factor, 2p+1, and double for each bit-depth up to the bit-depth you want. Finally, divide by 2 (eliminating potentials not meeting the 8n-1 or 8n+1 rule). So, the first potential factor of 727 would be 1455 (an 11-bit number). Take 1 for the number of potential 11-bit factors, and double over and over until you get where you want (2 12-bit potentials, 4 13-bit, 8 14-bit, etc.). However, 33,219,283's first potential factor is 66,438,567 (a 26-bit number). It has 1 26-bit potential, 2 27-bit, 4 28-bit, etc. Take these numbers and divide by 2 for the numbers of potential factors that need testing. You can see how there are *way* more potential 57-bit factors for 727 than 33,219,283. NOTE: These figures are approximate as 727 may only have say 3 potential 14-bit factors to test. (Actually, it only has 2!). It will give you a good idea of the number of potential factors you are looking at testing for any given bit-depth though... To illustrate better: To test the exponent 727 at 2^57 (only 57-bit factors), you must test approx. 7 x 10^13 potential factors. To test the exponent 33,219,283 at 2^57 (only 57-bit factors), you must only test approx. 2.15 x 10^9 potentail factors. BTW, there is a more accurate way to determine the exact number of potential factors that need to be tested at any given depth, but requires a little more effort. And when we're talking a difference of a couple million potential factors with a total of 100 trillion potential factors to test, is it really important to know the exact number?? Eric Hahn _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 18:12:38 -0700 From: Spike Jones <[EMAIL PROTECTED]> Subject: Mersenne: stopping a LL test > [ Joth Tupper explained] ...why we cannot stop in the "middle" > of a Lucas-Lehmer test. The essential answer is that we > know a property of particular terms in the sequence 4, 14, 192... > given recursively by squaring and subtracting 2.... I understand this now, but I didnt when I first downloaded and started GIMPS over a year ago, so heres a funny story for you GIMPSers to add to your dumb-spike collection: {8^D I downloaded GIMPS and started it thinking that it was doing a brute force factorization and assuming it would stop as soon as it found a factor. So my excitement built steadily as it passed 50% finished, then 60%, etc, and by the time it was in the 80s and still running, I was checking my computer every hour or so, cheering wildly each time {8^D haaa ha haaa, laughing maniacally, with my wife quite convinced I was going crazier, and by the time I was in the mid 90s I was beside myself, thinking I had perhaps nailed a mersenne prime on my very first try, and (blush) actually calculating the probability of this happening, etc. When my LL run hit 98 percent, I could *not* stop watching the computer screen, nor did I dare actually *use* the computer, expecting at any time a message like "bzzzt. 2^3123929-1 has a factor of yakkity yak". Since I am embarassing myself anyway, and recording a stupid-spike story in a medium far more permanent than any book or newspaper, I might as well admit that I stayed up until 2 am, on a work night, to see if I had indeed hit the jackpot on the very first try. When it hit 100% and reported: "2^3123929-1 is not prime", well, imagine my dismay. {8-[ {8^D Since then, I have learned all about the Lucas Lehmer algorithm, reignited my 15-years-dormant love of number theory, scanned the number theory websites and learned a ton of cool math stuff. Im in nerdvana here. So, thanks George. How about posting a picture of yourself so we can print it out and frame it? {8^D spike _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 23:04:42 -0400 (EDT) From: Darxus <[EMAIL PROTECTED]> Subject: Re: Mersenne: Re: [Windows net utilities] On Tue, 12 Oct 1999, Steinar H. Gunderson wrote: > On Mon, Oct 11, 1999 at 10:10:04PM +0100, Brian J. Beesley wrote: > >Windows users might care to try a nice program called CyberKit, which > >is freeware & does ping, traceroute & NS lookup (amongst other > >things). > > I don't know if Windows does `other things', but it certainly has > ping (ping) and traceroute (tracert). It lacks a decent `host', > though, but ping will do most of the work for you. If you're looking for UNIXish network related utilities to run under windows.. or actually anuthing UNIXish to run under windows, get cygwin (http://sourceware.cygnus.com/cygwin). Cygwin is basically a port of the unix API, and this package comes with a lot of the standard utilities. And a lot of stuff compiles under it out of the box.. or tarball. I look forward to an entire Linux distribution being ported to it :) They're working on porting the X server to it. __________________________________________________________________ PGP fingerprint = 03 5B 9B A0 16 33 91 2F A5 77 BC EE 43 71 98 D4 [EMAIL PROTECTED] / http://www.op.net/~darxus Join the Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 20:43:23 -0400 (EDT) From: Darxus <[EMAIL PROTECTED]> Subject: Mersenne: gzipped binary expansion of the largest prime Is there a copy (preferably unformatted plaintext) of the decimal expansion of the largest (2million-some digits) prime, gzipped or zipped or something, somewhere ? Anybody know how long it takes to calculate this using the calc program (which I'm currently compiling for this purpose) ? I intend to print it out in like, 2 or 3 point font & wallpaper my room with it :) Somewhere I have a printout of some number of millions of digits of pi... I really want a hardbound copy of, say, 10 million digits of pi, with the greek letter stamped on the front. Some small font size. About.. 5 1/2" x 8 1/2". I dunno, maybe this group is into numbers enough that we could actually pool our resources and get it done. Or maybe I'm the only one who'd be interested. ? I actually looked into printing services once. Yeah, I could get it printed & bound, but not hard bound, with the letter pi stamped on it. Bummer, calc didn't compile in my shell account. In file included from seed.c:53: /usr/include/sys/resource.h:25: field `ru_utime' has incomplete type /usr/include/sys/resource.h:26: field `ru_stime' has incomplete type seed.c: In function `pseudo_seed': seed.c:216: field `tp' has incomplete type seed.c:241: field `tp2' has incomplete type *** Error code 1 make: Fatal error: Command failed for target `seed.o' Oh well, hope my repaired Linux drive gets back soon.... __________________________________________________________________ PGP fingerprint = 03 5B 9B A0 16 33 91 2F A5 77 BC EE 43 71 98 D4 [EMAIL PROTECTED] / http://www.op.net/~darxus Join the Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 13:59:55 -0700 From: Paul Leyland <[EMAIL PROTECTED]> Subject: RE: Mersenne: Factoring numbers... > Anyway, still waiting to hear if ECM will, > eventually, find all factors or if it leaves "factors" in the range... The best way of thinking about this is that each curve at a given bound has a small but non-zero probability of finding a factor of a certain length, assuming that one exists. There are a very large number of curves available, and so the probability of missing the factor on *all* the curves can be made extremely low --- but non-zero. This handwaving approach is far from rigorous but it is an extremely good approximation of the truth. So yes, in principle it can leave factors undiscovered. In practice, it will find them within a reasonable time --- but depending on your luck that time could be short or it could be lengthy. A few examples from personal experience might be illustrative. The largest factor I've found with ECM was a 45-digit factor of 423*2^423+1 and was found in phase 1 with the curve bound set at 3 million. I estimate I had a roughly one in twenty thousand chance of finding that factor in the number of curves I ran. At the other extreme, I recently used MPQS to finish off a factorization of an 89-digit integer, only to find that one of the factors had 30 digits. I had already found a 33-digit factor of the number for which the C89 was the cofactor, but missed the P30. I estimate that I had a less than 1% chance of missing the smaller factor given the amount of ECM work I had done. In the first case I was lucky, and the second I was unlucky. That's the nature of ECM. Paul _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 22:23:46 -0500 From: Ken Kriesel <[EMAIL PROTECTED]> Subject: Re: Mersenne: splitting up 10m digit primes As I posted some days back; Anyone who wants to quit an exponent after investing a PII-400-month or more, please contact me, and we'll try to carry it on, using it for the QA effort. It could take some major bandwidth-minutes if more than a few exponents are quit, however. Ken At 04:15 PM 1999/10/12 EDT, [EMAIL PROTECTED] wrote: >Well, as completion time gets longer and longer, it becomes more likely that a user will give up in disgust. This could be after several months of work is already complete. > >How about an option when you hit "QUIT GIMPS" to upload your P and Q files to Primenet, so someone can at least finish the job? _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 22:29:38 -0500 From: Ken Kriesel <[EMAIL PROTECTED]> Subject: Mersenne: Vaxen & Intel At 12:03 PM 1999/10/12 -0400, Jud McCranie <[EMAIL PROTECTED]> wrote: >At 12:54 PM 10/12/99 +1000, Simon Burge wrote: >>"John R Pierce" wrote: >> >> > a year on one of these [a vax] wouldn't equal one day on a pentium-II. >> >>Probably a bit generous there even, given that older vaxen wouldn't >>have pipelined FPU's so you might get one result every 10 (or perhaps >>even 100) clocks, as opposed to one result almost every clock on modern >>hardware. > >In his book "Programming Pearls", Jon Bentley gives the figure of 570 >seconds for sorting 10,000 integers on a VAX-11/750. My PII-300 does it >1160 times faster and my C-400 is 1780 times faster. But that is comparing >working with integers instead of FP. Hmm, a vax3900 is definitely not slower than a 780. The original Vax was the 780, followed by the slightly slower 750, both designed in the 1970's. A VS2000 was a tiny, almost shoebox size vax, and those were about 780 speed. A VS3100-38 (circa 1990) was about 4 x a 780; a VS4000-60 was about 12 times; I think a 4000-90 was about 30 times. My recollection was that for multiple precision programs in integer in C, a VS3100-38 was about half the speed of an Intel 386-33 with cache, and a VS4000-60 was about the equal of a 486-33. What sort algorithm are those figures for? In what programming language? Which compiler? There are lies, damned lies, and statistics. Then there are benchmarks... Ken _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 23:33:47 -0400 From: Walt Mankowski <[EMAIL PROTECTED]> Subject: Re: probability of primeness (was: Re: Mersenne: splitting up 10m digit primes) On Tue, Oct 12, 1999 at 10:53:18PM -0400, Darxus wrote: > > I'm hoping what I have to say in this email might be important. > > On Tue, 12 Oct 1999, George Woltman wrote: > > > At 04:12 PM 10/12/99 -0400, you wrote: > > >> >And how is the probability of finding a prime calculated ? > > >> > > >> It is roughly how-far-factored-in-bits * 2 / exponent > > > > > >Okay.. what's "how-far-factored-in-bits" mean ? > > > > I think trial factoring is done to 2^68 for an exponent around 33 million. > > Thus your chance is 2 * 68 / 33000000. > > Okay, so as far as we know, each number is equally likely to be prime, and > this probability is just based on how much has already been tested ? No, they're saying the probability is based on how deeply they've tried to factor the number before trying the LL test. The more numbers N you rule out as potential factors of M, the more likely M is to be prime. Also, although there are an infinite number of primes, their density thins out considerably as they get large because there are so many more potential factors below them. This applies to primes in general; I don't know if it applies to Mersenne primes. _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 23:38:36 -0400 (EDT) From: Lucas Wiman <[EMAIL PROTECTED]> Subject: Re: probability of primeness (was: Re: Mersenne: splitting up 10m digit primes) > > I think trial factoring is done to 2^68 for an exponent around 33 million. > > Thus your chance is 2 * 68 / 33000000. > > Okay, so as far as we know, each number is equally likely to be prime, and > this probability is just based on how much has already been tested ? Umm, no. The probability that a number is prime is inversly proportional to the number of digits (or more precisely, the probability that a number on the interval [1,x] is prime is asomtotically 1/ln(x)). However, the probability that a number is prime increases with the amount that has been trial factored (without finding a factor). The precise estimation is based on Mertel's theorem. > Ugh... I apparently had bad math teachers, and GIMPS is really making me > feel it. I *really* wanna play with these numbers, but I feel > intellectually cripled. You certainly seem to have done a good job! You found the two big conjectures about Mersenne distribution. That is Curt Noll's (poorly defined) island theory (your pairs observation), and your observation about it being roughly exponetial (a conjecture due to Wagstaff, and others, though Wagstaff's has hueristic evidence to back it up). > I took the numbers from > http://www.utm.edu/research/primes/mersenne.shtml#known See http://www.utm.edu/research/primes/notes/faq/NextMersenne.html, as well as http://www.tasam.com/~lrwiman/FAQ-mers, Q4.2. > Does anybody see what I'm talking about ? Is there any significance to > this ? Has somebody already written extensive papers on this ? Yes, see above. Good work! - -Lucas _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 23:50:19 -0400 (EDT) From: Lucas Wiman <[EMAIL PROTECTED]> Subject: Re: Mersenne: gzipped binary expansion of the largest prime > Is there a copy (preferably unformatted plaintext) of the decimal > expansion of the largest (2million-some digits) prime, gzipped or zipped > or something, somewhere ? Yes! Landon Curt Noll has all the Mersenne primes on his website. http://reality.sgi.com/chongo/prime/merdigit/index.html#largest (Ack! I'm slipping, I had to look it up!) Also (if you want it in poster form) look at www.perfsci.com (though last I checked it was ~$30 US). > Bummer, calc didn't compile in my shell account. > > In file included from seed.c:53: > /usr/include/sys/resource.h:25: field `ru_utime' has incomplete type > /usr/include/sys/resource.h:26: field `ru_stime' has incomplete type > seed.c: In function `pseudo_seed': > seed.c:216: field `tp' has incomplete type > seed.c:241: field `tp2' has incomplete type > *** Error code 1 > make: Fatal error: Command failed for target `seed.o' > > Oh well, hope my repaired Linux drive gets back soon.... Hmm, you might want to sent this to Noll. He is big on getting Calc to compile without errors on darn-near everything. Also look at Pari/GP. It has binaries for windows! (It seems faster than Calc too!) ftp://megrez.math.u-bordeaux.fr/pub/pari/windows (take off the /windows for linux/msdos/mac/etc versions). Hope this helps, Lucas _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 12 Oct 1999 23:04:31 -0600 From: "Aaron Blosser" <[EMAIL PROTECTED]> Subject: RE: probability of primeness (was: Re: Mersenne: splitting up 10m digitprimes) > The chart ooked interesting. Approximately exponential. But it didn't fit > real cleanly. So I stared at it for a long time. > > I noticed that the last 6 mersenne primes appeared to be in pairs... <...> > Does anybody see what I'm talking about ? Is there any significance to > this ? Has somebody already written extensive papers on this ? > > I don't have access to a charting program this instant, or I'm sure I'd be > up the rest of the night playing with this. Hopefully I'll get some time > to do so tomorrow. I'm sure someone beat me to a reply already, but... This is what's known as Noll's Island Theorem, if I am remembering the name correctly (and someone is bound to correct me otherwise :-) A while back on this list, we had a discussion on just this very thing, but as it turns out, with some more notable examples, as you have seen, the deviation, in percent, between so called pairs is pretty far and wide. And then you have to ask, well, what constitutes a pair anyway? Within 10% of each other? 20%? Some pairs are ridiculously close when you look at 'em, and some are pretty far apart but still close enough that some would call it a pair still. And there's no real correlation to the size of the exponents either. The deviation grows and shrinks all along the list of primes. There does seem to be a general trend, but beyond that, nothing too notable. There are exceptions to the trend which makes me think it's just a case of we humans trying to place an order on something where no order can be found. We do the same thing when looking at clouds and we start to see shapes of actual things. Power of imagination is all, I suspect. But the only way to know for sure is to find more Mersenne Primes, so let's all keep looking! I'll take this opportunity to say this: Some of us have volunteered to pony up some dough to give to the next person who finds a Mersenne Prime. This goes back to the way it was *before* the EFF prize. Since it'll be a while before we find a 10M digit prime (unless someone gets REAL lucky), it's nice to have a small incentive for someone to find another one before that. Did we ever "elect" anyone to keep track of "donations"? Scott? George? :-) Once we figure that out, maybe a little pool could get going to add to the already pledged amounts. If you're really (and I mean *really*) excited about new discoveries, maybe you'd like to pledge $10 or $20 or however much. I've pledged more, but that's more of an atonement for things I did last year, besides the thrill of finding a new Mersenne prime. :-) Aaron _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 00:16:26 -0400 (EDT) From: Darxus <[EMAIL PROTECTED]> Subject: Mersenne: primes & bases Are prime numbers prime in all bases ? __________________________________________________________________ PGP fingerprint = 03 5B 9B A0 16 33 91 2F A5 77 BC EE 43 71 98 D4 [EMAIL PROTECTED] / http://www.op.net/~darxus Join the Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 14:24:31 -0400 From: Jud McCranie <[EMAIL PROTECTED]> Subject: Mersenne: Error 11 I just communicated with the server, and got error 11 - exponent already tested on one I'm 95% through with. (A few weeks ago I had to transfer some exponents from one machine to another, so something may have gotten mixed up in the process.) Will this still count as a double check of that exponent? I have some other exponents queued up (that I transferred) - would the communication with the server warn me if they have been tested? +---------------------------------------------------------+ | Jud McCranie | | | | Programming Achieved with Structure, Clarity, And Logic | +---------------------------------------------------------+ _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 10:44:04 -0400 From: Jeff Woods <[EMAIL PROTECTED]> Subject: Re: Mersenne: stopping a LL test At 06:12 PM 10/12/99 -0700, you wrote: >math stuff. Im in nerdvana here. So, thanks George. How >about posting a picture of yourself so we can print it >out and frame it? {8^D spike http://www.utm.edu/research/primes/bios/bio.html#Woltman _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 14:06:43 -0400 From: Jud McCranie <[EMAIL PROTECTED]> Subject: Re: Mersenne: primes & bases At 12:16 AM 10/13/99 -0400, Darxus wrote: >Are prime numbers prime in all bases ? Yes. The base of the number is just how we write it - it is not the number itself. +---------------------------------------------------------+ | Jud McCranie | | | | Programming Achieved with Structure, Clarity, And Logic | +---------------------------------------------------------+ _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 14:03:22 -0400 From: Jud McCranie <[EMAIL PROTECTED]> Subject: Re: Mersenne: Vaxen & Intel At 10:29 PM 10/12/99 -0500, Ken Kriesel wrote: > What sort algorithm are those figures for? In what programming language? >Which compiler? It was insertion sort, based on Bentley's pseudocode, so it was the same algorithm. I coded it in Pascal, he did it in C. I used Stony Brook Pascal+ ver 7.10, I don't know what compiler he used, but it was probably the standard C compiler for the VAX. >There are lies, damned lies, and statistics. Then there are benchmarks... There are some variables that are not accounted for. +---------------------------------------------------------+ | Jud McCranie | | | | Programming Achieved with Structure, Clarity, And Logic | +---------------------------------------------------------+ _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 13:24:22 -0700 From: "Joth Tupper" <[EMAIL PROTECTED]> Subject: Re: probability of primeness (was: Re: Mersenne: splitting up 10m digit primes) Some of this is likely to be based on the density of primes. The "Prime Number Theorem" shows that the asymptotic density of primes is x / ln x. This density is often written pi(x) [the lower case Greek letter, btw] with pi(x) = (the number of primes less than or equal to x) / x. This is not a trivial result and it has been over 30 years since I looked at it. There is an "elementary" proof [nothing more than calculus] that takes about 40 pages, as I recall. This spends a lot of time on Mobius inversions -- but the details are gone without going back to sources. [It might be just the number of primes less than x, but for large x this is pretty similar.] AFAIK, the question of the number of Mersenne primes is open. (That is: finite or infinite set.) Joth - ----- Original Message ----- From: Walt Mankowski <[EMAIL PROTECTED]> To: mersenne <[EMAIL PROTECTED]> Sent: Tuesday, October 12, 1999 8:33 PM Subject: Re: probability of primeness (was: Re: Mersenne: splitting up 10m digit primes) > On Tue, Oct 12, 1999 at 10:53:18PM -0400, Darxus wrote: > > > > I'm hoping what I have to say in this email might be important. > > > > On Tue, 12 Oct 1999, George Woltman wrote: > > > > > At 04:12 PM 10/12/99 -0400, you wrote: > > > >> >And how is the probability of finding a prime calculated ? > > > >> > > > >> It is roughly how-far-factored-in-bits * 2 / exponent > > > > > > > >Okay.. what's "how-far-factored-in-bits" mean ? > > > > > > I think trial factoring is done to 2^68 for an exponent around 33 million. > > > Thus your chance is 2 * 68 / 33000000. > > > > Okay, so as far as we know, each number is equally likely to be prime, and > > this probability is just based on how much has already been tested ? > > No, they're saying the probability is based on how deeply they've > tried to factor the number before trying the LL test. The more > numbers N you rule out as potential factors of M, the more likely M is > to be prime. > > Also, although there are an infinite number of primes, their density > thins out considerably as they get large because there are so many > more potential factors below them. This applies to primes in general; > I don't know if it applies to Mersenne primes. > _________________________________________________________________ > 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, 13 Oct 1999 11:10:29 -0400 (EDT) From: Darxus <[EMAIL PROTECTED]> Subject: Re: Mersenne: gzipped binary expansion of the largest prime Can somebody give me the last few digits of the decimal expansion of (2^6972593)-1 so that I can verify my copy's intact ? __________________________________________________________________ PGP fingerprint = 03 5B 9B A0 16 33 91 2F A5 77 BC EE 43 71 98 D4 [EMAIL PROTECTED] / http://www.op.net/~darxus Join the Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 10:16:30 -0400 (EDT) From: Darxus <[EMAIL PROTECTED]> Subject: Mersenne: cooperative prime number award question On Tue, 12 Oct 1999, Aaron Blosser wrote: > Some of us have volunteered to pony up some dough to give to the next person > who finds a Mersenne Prime. This goes back to the way it was *before* the > EFF prize. Since it'll be a while before we find a 10M digit prime (unless > someone gets REAL lucky), it's nice to have a small incentive for someone to > find another one before that. > > Did we ever "elect" anyone to keep track of "donations"? Scott? George? > :-) Once we figure that out, maybe a little pool could get going to add to > the already pledged amounts. If you're really (and I mean *really*) excited > about new discoveries, maybe you'd like to pledge $10 or $20 or however > much. I've pledged more, but that's more of an atonement for things I did > last year, besides the thrill of finding a new Mersenne prime. :-) I'd been thinking about this some. I'd be willing to put in at least $20 a year (after all, it is an ongoing thing) toward a prize of up to $5k for every prime found, reguardless of how it is found, as long as there is full disclosure -- similar to the rules of the cureent EFF contest. I think the EFF is in a good position to set up a fund, so I cc'd them in this email. I'm hoping that's appropriate. __________________________________________________________________ PGP fingerprint = 03 5B 9B A0 16 33 91 2F A5 77 BC EE 43 71 98 D4 [EMAIL PROTECTED] / http://www.op.net/~darxus Join the Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 10:16:47 -0400 (EDT) From: Darxus <[EMAIL PROTECTED]> Subject: Mersenne: cooperative prime number award question On Tue, 12 Oct 1999, Aaron Blosser wrote: > Some of us have volunteered to pony up some dough to give to the next person > who finds a Mersenne Prime. This goes back to the way it was *before* the > EFF prize. Since it'll be a while before we find a 10M digit prime (unless > someone gets REAL lucky), it's nice to have a small incentive for someone to > find another one before that. > > Did we ever "elect" anyone to keep track of "donations"? Scott? George? > :-) Once we figure that out, maybe a little pool could get going to add to > the already pledged amounts. If you're really (and I mean *really*) excited > about new discoveries, maybe you'd like to pledge $10 or $20 or however > much. I've pledged more, but that's more of an atonement for things I did > last year, besides the thrill of finding a new Mersenne prime. :-) I'd been thinking about this some. I'd be willing to put in at least $20 a year (after all, it is an ongoing thing) toward a prize of up to $5k for every prime found, reguardless of how it is found, as long as there is full disclosure -- similar to the rules of the cureent EFF contest. I think the EFF is in a good position to set up a fund, so I cc'd them in this email. I'm hoping that's appropriate. __________________________________________________________________ PGP fingerprint = 03 5B 9B A0 16 33 91 2F A5 77 BC EE 43 71 98 D4 [EMAIL PROTECTED] / http://www.op.net/~darxus Join the Great Internet Mersenne Prime Search http://www.mersenne.org/prime.htm _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 08:13:05 -0700 From: Paul Leyland <[EMAIL PROTECTED]> Subject: RE: Mersenne: primes & bases > Are prime numbers prime in all bases ? That is a deep question. If by "base" and "prime" you are restricting yourself to the integers, the answer is "yes". If you allow yourself more freedom and allow other numeric quantities as your "base", the answer is "not necessarily". For example, in the integers 5 is prime, no matter what integral base you choose. On the other hand, if your "base" is sqrt(5), 5 is a perfect square. Over the complex numbers, the factorization of 5 is (2+i)*(2-i). The generalization of the concept of "prime", "base", "number" and so forth is the starting point of a very large chunk of mathematics... Paul _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Wed, 13 Oct 1999 20:06:17 -0400 (EDT) From: Lucas Wiman <[EMAIL PROTECTED]> Subject: Re: Mersenne: gzipped binary expansion of the largest prime > Can somebody give me the last few digits of the decimal expansion of > (2^6972593)-1 so that I can verify my copy's intact ? to find the last n base-d digits of M39, find 2^(6972593) (mod 10^d) -1 The last 100 digits of it are: 854323570491331747687718276359853562553418155924593120827624505017498840034615135366526142924193791 - -Lucas _________________________________________________________________ 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 #642 ******************************
