Mersenne Digest Saturday, June 26 1999 Volume 01 : Number 588 ---------------------------------------------------------------------- Date: Thu, 24 Jun 1999 09:54:34 -0700 From: Paul Leyland <[EMAIL PROTECTED]> Subject: RE: Mersenne: safe to defrag? > From: Jud McCranie [mailto:[EMAIL PROTECTED]] > Since Prime95 writes to the disk periodically, is it safe to do a disk > defragmentation while it is running? That, of course, depends on how good your defragger is. Personally, I wouldn't use a defragger that can't be trusted to leave the disk usable, even if disk activity is occurring. I won't make any product recommendations and, despite my email address, the contents of this mail is entirely my personal view and not that of a certain software company. However, I will say that I use / have used a variety of defraggers and Prime95 concurrently on Win9x, NT 4.0 at various service pack levels, and Win2k beta 3. Your milage may vary. Paul ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Thu, 24 Jun 1999 13:12:49 EDT From: [EMAIL PROTECTED] Subject: Mersenne: Defragmenting and Security <<I have mine configured to be a tray icon, which generally goes unnoticed or is ignored.>> How about the No Icon option? (You can still access it by trying to run Prime95.exe again). And have it configured as a Win95 service. I'm not sure if my system is an anomaly, but even the Three-Fingered Salute doesn't show Prime95 to be on the list of tasks to shut down. If the weasel who stole your laptop doesn't look hard enough, he will then have almost no chance of finding the program. Heh. In regards to defragmenting: I have defragmented my C hard drive (where Prime95 runs) a number of times, and it almost always takes over 30 minutes, which is how long Prime95 waits before writing save files. MS Defrag (taken from Symantec - look at the copyright notice!) is just as well-behaved as good old FAT16 Symantec Norton Utilities' defragmenter. Apparently, when a program writes to the HD, MS Defrag detects it and rescans the hard drive's contents. No errors are produced. If you are using a Symantec product, then it will also be well behaved. I don't know about other programs or non-Win95 systems. STL ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Thu, 24 Jun 1999 14:36:28 -0400 From: "Silverman, Bob" <[EMAIL PROTECTED]> Subject: Mersenne: RE: Mersenne Digest V1 #587 Regarding the discussion about the distribution of M_p: Sam Wagstaff's results imply that the expected number of Mersenne primes between 2^h and 2^2h is exp(gamma). Thus, they DO get progressively rarer. Further, by the PNT, the probability that a random integer near x is prime is 1/log(x). *ASSUMING* that 2^p-1 behaves like a random integer, the probability that it is prime should be 1/p log(2). Now, sum from 2 to k of 1/p is asymptotically loglog k [this is easy; p_n ~ n log n from PNT, so by Stieltje's integration (or Euler-Maclauren) on gets sum from 2 to k of 1/p = integral from 2 to k of 1/(n log n) d [n]. Now integrate by parts. ] Thus, one should expect that the number of Mersenne primes up to k is O(log log k). Be wary of what Richard Guy calls the law of small numbers... Most number-theoretic phenomena only show their true behavior for VERY large numbers and we are not there yet. To put it another way, as John Selfridge said: although we know loglog n goes to infinity, it has never actually been observed to do so... BTW, there is nothing unique about base 2 in this regard. We should expect that the number of primes of the form (a^p - 1)/(a-1) up to k is O(log log k) for all a. The only thing that changes is the implied constant. Bob Silverman ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Thu, 24 Jun 1999 15:20:33 -0400 (EDT) From: lrwiman <[EMAIL PROTECTED]> Subject: RE: Mersenne: Once again factoring > Exactly. I knew that 1/p number didn't look right. Isn't it more like > 1/sumof(all p < current p)? The method that I used (I think I got it from _Primes and Programming_ by Peter Glibin) is that for any given set of primes, the probability that any number greater than them will be divisable by at least one of them is ~1-prod((p_i-1)/p_i). The probability that any given prime p will divide another (>=2*p) is 1/p. Therefore, the probability that it will not be divisable is (p-1)/p. Now to get the probability that the number will not be divisable by two primes p_1 and p_2, we multiply the individual probabilities that they do not divide =(p_1-1)*(p_2-1)/(p_1*p_2). Therefore to find the probaility that they do divide (since prob. of divisability +prob of nondivisability=1), we take 1-(prob that they do not divide)=1-(p_1-1)*(p_2-1)/(p_1*p_2), and so on for p_3,p_4....p_n. Note that this primes do not have to be in numerical order after a time, for suficiently many primes the only ones missed will be powers of the primes not included which are very insignificant statistically. We can also use this method as a slight alteration of Euclid's proof for an infinity of primes, which I leave as an excersize for the readers ;) - -Lucas Wiman P.S. This is not rigorously proven, and might not take into account certain things like numbers divisable by p_1*p_2*p_3 or something like that. ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Thu, 24 Jun 1999 16:55:59 -0400 From: Jud McCranie <[EMAIL PROTECTED]> Subject: Mersenne: Mersenne Distribution At 02:36 PM 6/24/99 -0400, Silverman, Bob wrote: >Regarding the discussion about the distribution of M_p: > >Sam Wagstaff's results imply that the expected number of >Mersenne primes between 2^h and 2^2h is exp(gamma). >Thus, they DO get progressively rarer. This sounds right. This corresponds to the c^n law, where c = e^(log(2)/e^gamma) = 1.47576, which is also in better agreement with the current data than c=3/2 (and 3/2 has no justification either). I'd put my money on Wagstaff's estimate. +----------------------------------------------+ | Jud "program first and think later" McCranie | +----------------------------------------------+ ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Thu, 24 Jun 1999 18:03:17 -0400 (EDT) From: lrwiman <[EMAIL PROTECTED]> Subject: Re: Mersenne: Mersenne Distribution > This sounds right. This corresponds to the c^n law, where c = > e^(log(2)/e^gamma) = 1.47576, which is also in better agreement with the > current data than c=3/2 (and 3/2 has no justification either). I'd put my > money on Wagstaff's estimate. This would put the 38th mersenne at ~2.6million, the 39th at ~3.9million, and the 40th at 5.7million. I don't know much about statistics, but this seems like a bad match. Could it be that the form should be of c^(a*n+b), and the first few values might be anomalies for some reason? - -Lucas Wiman ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Thu, 24 Jun 1999 18:04:35 -0400 From: Peter Doherty <[EMAIL PROTECTED]> Subject: RE: Mersenne: safe to defrag? At 09:54 06/24/1999 -0700, you wrote: >> From: Jud McCranie [mailto:[EMAIL PROTECTED]] > >> Since Prime95 writes to the disk periodically, is it safe to do a disk >> defragmentation while it is running? My computer is up 24/7. Prime95 writes to disk every 30 minutes, and my email program (Eudora) checks for new mail every 20 minutes. Three times a week my computer automatically runs Win98 defrag program to defrag all 7.5GB of HDD space (a 4.3 HDD, and a 3.2HDD) I've never had any real problem. Everytime something writes to disk defrag restarts, scans for errors, and then checks over all that it has defraged, and picks up where it left off. This takes 1 to 2 minutes. I've had some unexplained computer hangs while defragging, but I can't attribute that to Prime95. So that's the long. The short is, it's entirely safe to defrag while running Prime95. - --Peter ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Thu, 24 Jun 1999 15:44:44 -0700 From: Luke Welsh <[EMAIL PROTECTED]> Subject: Re: Mersenne: Date: Thu, 24 Jun 1999 09:01:14 -0600 At 08:02 AM 6/24/99 -0700, you wrote: > ERROR: Primenet error: 12029 > >Can someone help me with the meaning of this particular >message. See http://entropia.com/ips/faq.html - --Luke ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Thu, 24 Jun 1999 22:25:28 -0400 From: Jud McCranie <[EMAIL PROTECTED]> Subject: Re: Mersenne: Mersenne Distribution At 06:03 PM 6/24/99 -0400, lrwiman wrote: >This would put the 38th mersenne at ~2.6million, the 39th at ~3.9million, >and the 40th at 5.7million. It doesn't work quite that way. It is a global property that doesn't say anything about the individual Mersenne primes, just as the Prime Number Theorem applies to the distribution of primes as a whole, but doesn't tell you about individual primes. +----------------------------------------------+ | Jud "program first and think later" McCranie | +----------------------------------------------+ ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Fri, 25 Jun 1999 12:04:52 -0400 From: "Ernst W. Mayer" <[EMAIL PROTECTED]> Subject: Mersenne: Ernst's CA or bust deal (Was: M#38) Dear Mersenners: This being the sesquicentennial of the California gold rush, I thought a fitting tribute would be to move there, this time to be nearer the silicon, rather than the gold. I was going to insert an Au-ful pun here, but couldn't come up with one. Si-gh... As of 1 july, my left coast address is Ernst W. Mayer 10190 Parkwood #1 Cupertino, CA 95014 And of course you can always use my "permanent" E-address, <[EMAIL PROTECTED]>. I'd like to thank George W. for giving me the opportunity to verify M#38, (I don't normally do double-checking, but in this case decided to make an exception :) and David Willmore for use of his employer's Alpha 21264 for the two-week-long run. I won't have a chance to finish the production version of the prototype code David used for his run (about 20% faster than my current production Mlucas code) until I'm settled into my new apartment, but that'll give you all time to order your own 21264's (about $7500 per CPU via microway.com, but perhaps GIMPSers can get a volume discount - at the above price, it's still far cheaper than an equivalent MIPS). And I see that the movers are here, so I must prepare to take nigel off-line. Cheers, Ernst ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Sat, 26 Jun 1999 02:37:51 -0400 (EDT) From: Lucas Wiman <[EMAIL PROTECTED]> Subject: Mersenne: Still more 10,000,000+ digit factors!!!!!!! I have found 1868 new factors in the range of Brian's 10,000,000+ digits. All of the other primes in this range have been tested through 2^47. They are avalaible at: http://www.tasam.com/~lrwiman/fact47 or http://www.tasam.com/~lrwiman/fact47.gz - -Lucas Wiman P.S. If these posts are getting annoying, at least they will be getting less frequent. P.P.S. Does anybody but Will care about these new factors? ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Sat, 26 Jun 1999 09:26:52 -0500 (CDT) From: Conrad Curry <[EMAIL PROTECTED]> Subject: Re: Mersenne: Once again factoring On 23 Jun 99, at 6:17, Brian J. Beesley wrote: > > On 22 Jun 99, at 17:38, Gary Diehl wrote: > > 2. Why use a table at all? Is it faster than doing a calculation to > > determine if [f % 255255] != 0 ? (I know sometimes tables can speed > > things up, but does it really help with so few numbers involved in the > > table?) Calculating f % 255255 does not tell us if f is divisible by 3, 5, 7, 11, 13 or 17 but only their product. You would want to test if GCD (f, 255255) > 1. > On a Pentium, recalculating the index & looking up the table takes of > the order of 20 clocks (allowing for a cache miss on the table read), > irrespective of the size of f, whereas just dividing f by 255255 to > get the remainder costs 39 clocks, even if f/255255 < 2^32. If f = 2*k*P + 1 is a potential factor of 2^P-1, then we can calculate s[i] == -(p[i] + 1)/2 (mod p[i]) where p[i] are the primes 3, 5, 7, ... Then solve the linear congruence P * c[i] == s[i] (mod p[i]) for c[i]. To test if f is divisible by p[i] test if k == c[i] (mod p[i]). That reduces the dividend from f to k. The division can be done by multiplication by the reciprocal of p[i], this and c[i] can be precomputed. ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ Date: Sat, 26 Jun 1999 13:32:26 -0400 From: Jud McCranie <[EMAIL PROTECTED]> Subject: Mersenne: Distribution of Mersenne primes For those of us who don't have access to Wagstaff's 1983 paper "Divisors of Mersenne Numbers", it is nicely summarized in "The New Book of Prime Number Records", by Paulo Ribenboim, chapter 6, section V.A. (page 411-413 in this edition). He gives 3 statements: (a) The number of Mersenne primes < x is about log(log(x))*e^gamma/log(2) (b) the expected number of Mersenne primes between x and 2x is about e^gamma. (equivalent to part a) (c) the probability that Mq is prime is about c*log(aq)/q where c=e^gamma/log(2) and a=2 if q = 1 mod 4; a=6 if q=1 mod 4. It gives fours considerations upon which Wagstaff's conjecture is based. Of course, these imply that the nth Mersenne number is about [2^(-gamma)]^n, or 1.4759^n. He goes on to mention Eberhart's earlier conjecture of (3/2)^n, but states that there is no serious reason supporting this version of the conjecture. +----------------------------------------------+ | Jud "program first and think later" McCranie | +----------------------------------------------+ ________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm ------------------------------ End of Mersenne Digest V1 #588 ******************************
