Mersenne Digest Sunday, June 27 1999 Volume 01 : Number 589
----------------------------------------------------------------------
Date: Sat, 26 Jun 1999 13:59:15 -0400
From: "Geoffrey Faivre-Malloy" <[EMAIL PROTECTED]>
Subject: Mersenne: Another factor found for Fermat 16
I found another factor for Fermat 16. What do I do now? How can I factor
this number that I found? Are there programs out there that will let me do
that?
FYI, the factor is:
M16384 has a factor:
3178457030898592746194675570663374420833971377365687459461386297851584459031
8073180374859604847822828243686877928403667633015295
G-Man
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Date: Sat, 26 Jun 1999 15:58:33 -0400
From: Allan Menezes <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Distribution of Mersenne primes
According to Paulo Ribenboim's book quoted below by Jud Euler's Constant
gamma=0.577215665... and working out the number of mersenne primes below p=7000000
in Mathematica 4.0 gives 39.5572 primes, so we must be missing a prime if
Wagstaffs' right.
Allan Menezes
Jud McCranie wrote:
> 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
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Date: Sat, 26 Jun 1999 23:15:57 +0200
From: "Steinar H. Gunderson" <[EMAIL PROTECTED]>
Subject: Mersenne: Re: [Fermat]
Here is the complete factorization of your number, directly from giantint.
As before, some of these factors may be composite.
3
* 5
* 17
* 257
* 641
* 65537
* 87596535553
* 12360473009170367279616001
* 6700417
* 26017793
* 63766529
* 190274191361
* 67280421310721
* 1256132134125569
* 59649589127497217
/* Steinar */
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Date: Sat, 26 Jun 1999 21:33:59 GMT
From: [EMAIL PROTECTED] (Steven Whitaker)
Subject: Re: Mersenne: Another factor found for Fermat 16
On Sat, 26 Jun 1999 13:59:15 -0400, you wrote:
>I found another factor for Fermat 16. What do I do now? How can I factor
>this number that I found? Are there programs out there that will let me do
>that?
>
>FYI, the factor is:
>
>M16384 has a factor:
>3178457030898592746194675570663374420833971377365687459461386297851584459031
>8073180374859604847822828243686877928403667633015295
>
>G-Man
G-Man,
I'm afraid I have some bad news. M16384 is not a Fermat number. It is
2^(2^14)-1 whereas a Fermat number is of the form 2^(2^n)+1. [Note the
signs]. If you want to factor Fermat numbers, try P16384.
However, all may not be lost. We can factorise M16384 algebraically as
(2^8192+1)(2^4096+1)(2^2048+1)...(2^4+1)(2^2+1)(2^1+1) [ie
F13*F12*F11...*F2*F1*F0] That means that if we remove all the known
factors of these Fermat numbers from your factor, we may be left with
a residue greater than 1. That must be a previously unknown factor
either of F12 or F13 (all the others having only known factors).
Unfortunately, as I'm not a mathematician, I don't have a factoring
program that can deal with numbers of that size - but I'm sure some of
the real mathematicians on the list can oblige. If you want a factorer
that can handle somewhat smaller numbers, try factor.exe. You can
download this from Chris Caldwell's prime pages
(http://www.utm.edu/research/primes/index.html) which also contain a
lot of other interesting prime-related information.
Best of luck
Steve
29 Mersenne factors found and counting
Well, if you do find a factor of a Fermat number,
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Date: Sat, 26 Jun 1999 19:33:59 -0400 (EDT)
From: Lucas Wiman <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Still more 10,000,000+ digit factors!!!!!!!
>> P.P.S. Does anybody but Will care about these new factors?
>Heck yeah... keep 'em coming... a bit more discussion of how you're
>finding these things could be interesting, tho...
Well, that is probably the least interesting part. I use Will's
MersFacGMP program on a PII/233 running RedHat 5.2.
> Factoring to 2^47 is fairly easy, I suppose, but where are all these
> stragglers coming from?
Well yes, especially when the numbers are as big as 33mil, but when there
are 91423 remaining numbers, it takes a bit longer to calculate.
(89555 after factoring to 2^47). Now I'm trying 2^50, (that should take a
while). What do you mean by stragglers?
- -Lucas Wiman
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Date: Sat, 26 Jun 1999 19:59:25 -0400 (EDT)
From: "Vincent J. Mooney Jr." <[EMAIL PROTECTED]>
Subject: Mersenne: Factoring and Databases
I may be a little obtuse here (and spelling, expression of ideas may be
inadequate) but....
A Mersenne number's prime divisors are unique to that number. Letting a and
b be primes, 2^a - 1 and 2^b - 1 have completely different factors. So we
can make a table (database) with
p1 divides M(q1)
p2 divides M(q2)
p3 cannot divide any Mersenne number
p4 divides M(q3)
p5 cannot divide any Mersenne number
p6 divides M(q2)
etc.
where p1 = 3, p2 = 5 and so on for all the primes in sequence.
The q1, q2 will also be primes but not necessarily in sequence and may occur
several times (as it does for p6). I will use the language that "pn belongs
to qm" to mean that the nth prime is a divisor of the Mersenne number q
sub-m (2^qm - 1). I can then say that pn (the nth prime) either fits
condition A (it divides M(qm) where qm is known) or that pn fits condition B
(it cannot divide any Mersenne number).
ALERT: Am I right about conditions A and B or does condition B work for
specific Mersenne numbers only?
I suspect that if someone looked at the primes up to p200, the 200th prime,
all have an associated Mersenne number M(qm) (M of q sub-m).
Lucas Wiman, for example, states that he has found 1868 new factors in the
range of Brian's 10,000,000+ digits, which I take to mean that 1,868 new
primes are known to belong to a specific qm.
Why test factor for primes in the range 2^1 to 2^10? If someone made the
table I described, it is possible that all primes less than 2^10 are in the
table I have described because they are known divisors of a Mersenne number
OR are not candidates for dividing any Mersenne number by other conditions
(subject to the ALERT above).
Therefore factoring could start at 2^10 + 1 and continue up to 2^x (where x
is a suitable integer that GIMPS choses).
Eventually, all primes in the range 2^10 to 2^11 will be covered (not a
candidate for a divisor of a Mersenne number or the prime is known to belong
to qm for some m). Then 2^11 from 2^12 will be covered, etc.
There is a key idea here, that each pn will eventually belong to some qm,
and that the smallest pn without a qm will rise over time due to the effort
of factorers such as Lucas Wiman. Would we be able to try brute force
factoring at some point for the Mersenne number M(s) where s > 10,000,000
since there are so many primes taken out of the potential list?
ALERT: I said "each pn will eventually belong to some qm". Is it possible
that p1234 (the 1,234th prime) is never a divisor of a Mersenne number?
O.K. Ken, Luke, Brian and you other smarter-than-me guys and good
mathematicians, fix this up, shoot it down, use it or whatever else is
possible.
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Date: Sat, 26 Jun 1999 21:11:21 -0400 (EDT)
From: Lucas Wiman <[EMAIL PROTECTED]>
Subject: re: Mersenne: Factoring and Databases
> Why test factor for primes in the range 2^1 to 2^10? If someone made the
> table I described, it is possible that all primes less than 2^10 are in the
> table I have described because they are known divisors of a Mersenne number
> OR are not candidates for dividing any Mersenne number by other conditions
> (subject to the ALERT above).
Remember that the smallest factor of M_p (p prime) is 2*p+1. This is
the smallest factor in a fair number of cases. This means, however,
that the smallest M_p which has a factor around 2^10 must have p~=2^9.
I think we're well past that point.
If I understand you correctly, you suggest making a massive table
of primes, and what M_p they divide. This would be very inefficient!!
Since I doubt that you could Store such a table in memory, the lookup
time would be enormous compared to the time spent to actually testing
the factor. Even if you could store it in memory, I think that the
lookup time would still be greater than the time spent testing the
factor.
If you mean to create a range (eg: <2^32) and test for which M_p divide
primes in that range one by testing the potential factors, this might
work, though Chris Nash did this on smaller ranges, and most of
the results weren't worth much. I think it is much faster to test exponents
for factors rather than the other way around, when dealing only with
prime exponents.
- -Lucas Wiman
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Date: Sat, 26 Jun 1999 22:01:43 -0400
From: Jud McCranie <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Distribution of Mersenne primes
At 03:58 PM 6/26/99 -0400, Allan Menezes wrote:
>According to Paulo Ribenboim's book quoted below by Jud Euler's Constant
>gamma=0.577215665... and working out the number of mersenne primes below
>p=7000000
>in Mathematica 4.0 gives 39.5572 primes, so we must be missing a prime if
>Wagstaffs' right.
It doesn't work that way. It is a global property about the average number of
Mersenne primes, so it doesn't say anything about a particular prime.
+----------------------------------------------+
| Jud "program first and think later" McCranie |
+----------------------------------------------+
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Date: Sat, 26 Jun 1999 21:21:05 -0700
From: Will Edgington <[EMAIL PROTECTED]>
Subject: Mersenne: Another factor found for Fermat 16
Geoffrey Faivre-Malloy writes:
I found another factor for Fermat 16. What do I do now? How can I
factor this number that I found? Are there programs out there that
will let me do that?
Yes, there are such programs. One is ecmfactor, a program I maintain
as part of the mers package at:
http://www.garlic.com/~wedgingt/mers.tar.gz
mers.tgz
(two file names with the same contents). Ecmfactor uses the Elliptic
Curve Method algorithm in the freeLIP library (written in C by Arjen
Lenstra and presently maintained by Paul Leyland). If freeLIP
compiles using your compiler, than ecmfactor should run fine.
FYI, the factor is:
M16384 has a factor:
3178457030898592746194675570663374420833971377365687459461386297851584459031
8073180374859604847822828243686877928403667633015295
I've appended the output from ecmfactor on this number.
Also, Steven Whitaker is quite correct; M16384 is, usually at least,
used to denote 2^16384 - 1, which is a Mersenne number. Fermat
numbers are of the form 2^(2^n) + 1. However, since 16384 is a power
of two, this particular Mersenne number, as Steven also notes, is
related to the Fermat numbers, since:
2^(2*x) - 1 = (2^x)^2 - 1^2 = (2^x - 1)*(2^x + 1)
(using the usual difference of squares method to get the two factors).
Since 16384 is a power of two, this can be repeated several times on
the 2^x - 1 half to get the factors that Steven lists.
And, if you at all interested in prime numbers and factoring, then
I strongly second Steven's suggestion of reading Chris Caldwell's
web pages at:
http://www.utm.edu/research/primes/index.html
Will
Ecmfactor says:
% ecmfactor -d < M16384.factor
prime factor: 3
prime factor: 5
prime factor: 17
prime factor: 257
new cofactor is composite
factor: 641
prime factor: 641
new cofactor is composite
factor: 114689
prime factor: 114689
new cofactor is composite
factor: 4179067011073
factor of factor: 65537
prime factor: 63766529
new cofactor is composite
factor: 26017793
prime factor: 26017793
new cofactor is composite
factor: 274177
prime factor: 274177
new cofactor is composite
factor: 974849
prime factor: 974849
new cofactor is composite
factor: 319489
prime factor: 319489
new cofactor is composite
factor: 6700417
prime factor: 6700417
new cofactor is composite
factor: 2424833
prime factor: 2424833
new cofactor is composite
tried 1 curves at bound 100 without success
tried 1 curves at bound 105 without success
tried 1 curves at bound 110 without success
tried 1 curves at bound 115 without success
tried 1 curves at bound 120 without success
tried 1 curves at bound 125 without success
tried 1 curves at bound 130 without success
factor: 45592577
prime factor: 45592577
new cofactor is composite
tried 1 curves at bound 135 without success
tried 1 curves at bound 140 without success
tried 1 curves at bound 145 without success
tried 1 curves at bound 150 without success
tried 1 curves at bound 155 without success
tried 1 curves at bound 161 without success
tried 1 curves at bound 167 without success
tried 1 curves at bound 173 without success
tried 1 curves at bound 179 without success
tried 1 curves at bound 185 without success
tried 1 curves at bound 191 without success
tried 1 curves at bound 197 without success
tried 1 curves at bound 203 without success
tried 1 curves at bound 209 without success
tried 1 curves at bound 215 without success
tried 1 curves at bound 221 without success
tried 1 curves at bound 227 without success
tried 1 curves at bound 233 without success
tried 1 curves at bound 239 without success
tried 1 curves at bound 245 without success
tried 1 curves at bound 252 without success
tried 1 curves at bound 259 without success
tried 1 curves at bound 266 without success
tried 1 curves at bound 273 without success
tried 1 curves at bound 280 without success
tried 1 curves at bound 287 without success
tried 1 curves at bound 294 without success
tried 1 curves at bound 301 without success
tried 1 curves at bound 308 without success
tried 1 curves at bound 315 without success
tried 1 curves at bound 322 without success
tried 1 curves at bound 329 without success
tried 1 curves at bound 336 without success
tried 1 curves at bound 343 without success
tried 1 curves at bound 350 without success
tried 1 curves at bound 358 without success
factor: 190274191361
prime factor: 190274191361
new cofactor is composite
tried 1 curves at bound 366 without success
tried 1 curves at bound 374 without success
tried 1 curves at bound 382 without success
tried 1 curves at bound 390 without success
tried 1 curves at bound 398 without success
tried 1 curves at bound 406 without success
tried 1 curves at bound 414 without success
tried 1 curves at bound 422 without success
tried 1 curves at bound 430 without success
tried 1 curves at bound 438 without success
tried 1 curves at bound 446 without success
tried 1 curves at bound 454 without success
tried 1 curves at bound 462 without success
tried 1 curves at bound 470 without success
tried 1 curves at bound 478 without success
tried 1 curves at bound 486 without success
tried 1 curves at bound 495 without success
tried 1 curves at bound 504 without success
tried 1 curves at bound 513 without success
tried 1 curves at bound 522 without success
tried 1 curves at bound 531 without success
tried 1 curves at bound 540 without success
tried 1 curves at bound 549 without success
tried 1 curves at bound 558 without success
tried 1 curves at bound 567 without success
tried 1 curves at bound 576 without success
tried 1 curves at bound 585 without success
tried 1 curves at bound 594 without success
tried 1 curves at bound 603 without success
tried 1 curves at bound 612 without success
tried 1 curves at bound 621 without success
tried 1 curves at bound 630 without success
tried 1 curves at bound 639 without success
tried 1 curves at bound 649 without success
tried 1 curves at bound 659 without success
tried 1 curves at bound 669 without success
tried 1 curves at bound 679 without success
tried 1 curves at bound 689 without success
tried 1 curves at bound 699 without success
tried 1 curves at bound 709 without success
tried 1 curves at bound 719 without success
tried 1 curves at bound 729 without success
tried 1 curves at bound 739 without success
tried 1 curves at bound 749 without success
tried 1 curves at bound 759 without success
tried 1 curves at bound 769 without success
tried 1 curves at bound 779 without success
tried 1 curves at bound 789 without success
tried 1 curves at bound 799 without success
tried 1 curves at bound 809 without success
tried 1 curves at bound 819 without success
tried 1 curves at bound 830 without success
tried 1 curves at bound 841 without success
tried 1 curves at bound 852 without success
tried 1 curves at bound 863 without success
tried 1 curves at bound 874 without success
tried 1 curves at bound 885 without success
tried 1 curves at bound 896 without success
tried 1 curves at bound 907 without success
tried 1 curves at bound 918 without success
tried 1 curves at bound 929 without success
tried 1 curves at bound 940 without success
tried 1 curves at bound 951 without success
tried 1 curves at bound 962 without success
tried 1 curves at bound 973 without success
tried 1 curves at bound 984 without success
tried 1 curves at bound 995 without success
tried 1 curves at bound 1006 without success
tried 1 curves at bound 1017 without success
tried 1 curves at bound 1028 without success
tried 1 curves at bound 1039 without success
tried 1 curves at bound 1051 without success
factor: 1256132134125569
prime factor: 1256132134125569
new cofactor is composite
tried 1 curves at bound 1063 without success
tried 1 curves at bound 1075 without success
tried 1 curves at bound 1087 without success
tried 1 curves at bound 1099 without success
tried 1 curves at bound 1111 without success
tried 1 curves at bound 1123 without success
tried 1 curves at bound 1135 without success
tried 1 curves at bound 1147 without success
tried 1 curves at bound 1159 without success
tried 1 curves at bound 1171 without success
tried 1 curves at bound 1183 without success
tried 1 curves at bound 1195 without success
tried 1 curves at bound 1207 without success
tried 1 curves at bound 1219 without success
tried 1 curves at bound 1231 without success
tried 1 curves at bound 1243 without success
tried 1 curves at bound 1255 without success
tried 1 curves at bound 1267 without success
tried 1 curves at bound 1279 without success
tried 1 curves at bound 1292 without success
tried 1 curves at bound 1305 without success
tried 1 curves at bound 1318 without success
tried 1 curves at bound 1331 without success
tried 1 curves at bound 1344 without success
tried 1 curves at bound 1357 without success
tried 1 curves at bound 1370 without success
factor: 67280421310721
prime factor: 67280421310721
cofactor is prime or a strong pseudo-prime
cofactor: 59649589127497217
So some of the factors that Steinar listed are composite, as he noted
was likely. All the numbers printed as 'prime factor's by ecmfactor
_might_ be merely strong pseudo-primes, to base-3 and four other,
random, bases greater than 2, but the odds are very much in favor of
them each being prime.
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Date: Sat, 26 Jun 1999 21:59:19 -0700
From: Will Edgington <[EMAIL PROTECTED]>
Subject: Mersenne: Another factor found for Fermat 16
Geoffrey Faivre-Malloy writes:
M16384 has a factor:
3178457030898592746194675570663374420833971377365687459461386297851584459031
8073180374859604847822828243686877928403667633015295
Further, if you try to divide this into M8192 (2^8192 - 1), you should
find that it factors that as well. That is, I checked all the factors
of the number above that my run of ecmfactor printed and they all
divided M8192 or an even smaller Mersenne (and therefore M16384,
M32768, etc).
The factoredM.txt and lowM.txt files in mersdata.tgz on my web pages
list, among other factors:
M( 256 )C: 59649589127497217
M( 512 )C: 1238926361552897
M( 1024 )C: 2424833
M( 1024 )C: 7455602825647884208337395736200454918783366342657
M( 2048 )C: 45592577
M( 2048 )C: 6487031809
M( 2048 )C: 4659775785220018543264560743076778192897
To here, the Mersenne numbers are completely factored (and are
therefore in factoredM.txt); from here on, the other (presumably
larger) factors are not known.
M( 4096 )C: 319489
M( 4096 )C: 974849
M( 8192 )C: 114689
M( 8192 )C: 26017793
M( 8192 )C: 63766529
M( 8192 )C: 190274191361
M( 8192 )C: 1256132134125569
M( 16384 )C: 2710954639361
M( 16384 )C: 2663848877152141313
M( 16384 )C: 3603109844542291969
M( 16384 )C: 319546020820551643220672513
My data contains no factors of M(32768) = M(2^15) or higher that are
not also factors of a smaller Mersenne number.
The ecm3 program of the mers package knows that composite exponent
Mersenne numbers have factors equal to smaller Mersenne numbers and
pulls all those factors out using repeated GCD's before trying to
factor the Mersenne number it is told to factor. E.g., it should
never print 319489 as a factor of M(8192) or any larger Mersenne,
since 319489 factors M(4096) but it will print 319489 as a factor of
M(4096) because it factors no smaller Mersenne number.
Will
http://www.garlic.com/~wedgingt/mersdata.tgz
http://www.garlic.com/~wedgingt/mers.tgz
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Date: Sat, 26 Jun 1999 22:23:19 -0700
From: Will Edgington <[EMAIL PROTECTED]>
Subject: Mersenne: Defragmenting and Security
[EMAIL PROTECTED] writes:
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.
I've noticed that my mother's Win98 machine also does not list Prime95
in the Startup menu, presumably because it's running as a Win98
service. But it does show briefly as a very small window in the lower
left hand corner of the screen during reboots while it's asking for
her password.
If you were to mark it as a hidden file as far as MSDOS is concerned
and put it in the Windows directory or a subdir thereof, it would be
even harder to find.
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.
My mother's machine also runs Prime95 24/7 and defrags automatically
every other week via the task scheduler. No problems yet of any kind.
Seti@Home, on the other hand, is not only a bigger resource hog, it
also interferes with Iomega's Zip backup program, causing backups to
fail by crashing the program right at the end, just before the backup
is complete. As I joked to her, I guess E.T. doesn't want her system
backed up.:)
Prime95, of course, doesn't interfere with backups.
Will
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Date: Sun, 27 Jun 1999 09:18:13 -0400
From: George Woltman <[EMAIL PROTECTED]>
Subject: Mersenne: Linux VFAT question
Hi,
Jan Scheller is having a problem (see below). Are there any Linux users that
can provide help? Please include Jan in your reply as Jan is not
subscribed to this mailing list.
I've already recommended the "poor man's" solution of
having mprime not write to disk every 30 minutes.
Best regards
George
<snip>
But the fact is that the performance of my mounted hdd win98 (vfat)
partition is very low. All my normal linux (ext2) partitions work with
normal performance. If I execute a command like `ls -l /Win98` (/Win98
is the path of may win98 (vfat) partition) it takes more than a second
to get the result. The second points is, that the swapping seems to be
slower than without running mprime. I run mprime every time at the
lowest priority.
Is there possibility to rise the performance of my vfat partition while
running mprime?
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Date: Sun, 27 Jun 1999 11:45:37 -0400
From: Pierre Abbat <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Linux VFAT question
> But the fact is that the performance of my mounted hdd win98 (vfat)
> partition is very low. All my normal linux (ext2) partitions work with
> normal performance. If I execute a command like `ls -l /Win98` (/Win98
> is the path of may win98 (vfat) partition) it takes more than a second
> to get the result. The second points is, that the swapping seems to be
> slower than without running mprime. I run mprime every time at the
> lowest priority.
cd to various directories, then type vdir|wc -l in each. This tells you how
many files the directory has. The more files, the longer it takes to sort them.
vdir -U lists the directory without sorting it, which is a little faster.
mprime just writes once every 30 minutes, so whether it's running in the
Windows drive or the Linux drive is inconsequential. As to the swapping, the
swap daemon runs at 12 naughty, so I don't see why mprime at 20 nice would
bother it at all.
phma
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Date: Sun, 27 Jun 1999 12:14:36 -0400
From: "Geoffrey Faivre-Malloy" <[EMAIL PROTECTED]>
Subject: Mersenne: A few questions
How large will the exponent be for a 10,000,000 digit prime number?
Has the prime number that was found a week ago been announced on this list?
I.E. What number was it?
Slovinski used to test huge numbers of primes...is he still doing that?
G-Man
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Date: Sun, 27 Jun 1999 12:58:06 -0400
From: Jud McCranie <[EMAIL PROTECTED]>
Subject: Re: Mersenne: A few questions
At 12:14 PM 6/27/99 -0400, Geoffrey Faivre-Malloy wrote:
>How large will the exponent be for a 10,000,000 digit prime number?
digits x 3.32192 gives the approximate exponent.
+----------------------------------------------+
| Jud "program first and think later" McCranie |
+----------------------------------------------+
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Date: Sun, 27 Jun 1999 13:34:46 -0400
From: "Geoffrey Faivre-Malloy" <[EMAIL PROTECTED]>
Subject: Mersenne: Statistics
There used to be a web site that showed top 100 type statistics and allowed
searching on the primenet results. Does anyone have that link?
G-Man
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Date: Sun, 27 Jun 1999 16:02:44 -0400
From: Jeff Woods <[EMAIL PROTECTED]>
Subject: Re: Mersenne: A few questions
At 12:14 PM 6/27/99 -0400, you wrote:
>How large will the exponent be for a 10,000,000 digit prime number?
P will be over 33,000,000 or roughly thereabouts. I'm sure someone can
give a more precise number.
>Has the prime number that was found a week ago been announced on this list?
>I.E. What number was it?
George has announced that the number that was suspected prime has been
confirmed to be prime by different hardware and algorithms. Because of
the EFF award money rules, we "GIMPS at large" will not be permitted to
know what the exponent was, or who discovered it, until an article has been
published in a peer-review academic research magazine or such.
George has said he is working on getting such a publication, but that no
publication date has been set. If he were to announce the prime NOW,
prior to publication, however, it COULD void the discoverer's claim to the
prize money, so I understand the caution in talking about it.
I suspect that were we to find another one between 1MM and 9.9MM digits,
where no prize money was at stake, that the announcement would be more
along the lines of M37 -- Slowinski or someone else confirming it, and just
getting it published in a newspaper. George has frequently used the SJ
Mercury News for that (in our last three discoveries). With the prize
money at stake, it is different.
>Slovinski used to test huge numbers of primes...is he still doing that?
While I don't know him from Adam, David SLOWINSKI (with a w) still works at
Cray, and still uses whatever spare CPU time he can locate to hunt for
primes.
I *suspect* that in light of GIMPS' success, he is likely looking much
higher than we are now (and has been for some time, again as a guess). He
knows our current program cannot top P > 20,000,000, so I suspect he's
above that range, perhaps by a good margin. It may be that he breaks GIMPS
record(s) again someday. I do know that the last time George expanded
Prime95 to hunt up to 20MM (up from 3MM), that George sent several residues
in the upper ranges to David for confirmation, and that David was able to
confirm SOME of them rather quickly (faster than a Cray could do the
calculations on the spot, so they were already tested). This to me
indicates that David is searching the numbers far above our top potential
range, especially since a Cray can test such numbers in about a week or
two, as a guess.
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Date: Sun, 27 Jun 1999 21:27:07 +0200
From: "Steinar H. Gunderson" <[EMAIL PROTECTED]>
Subject: Re: Mersenne: Linux VFAT question
On Sun, Jun 27, 1999 at 09:18:13AM -0400, George Woltman wrote:
>If I execute a command like `ls -l /Win98` (/Win98
>is the path of may win98 (vfat) partition) it takes more than a second
>to get the result. The second points is, that the swapping seems to be
>slower than without running mprime. I run mprime every time at the
>lowest priority.
Hmmm... I'm not completely sure if I understand the problem. Do you mean
that your vfat access is slower in general when running mprime? Or just
when you just have started mprime, and it's reading the p-file from
disk? And what is this `swapping' you're talking about, is that disk
access?
(A simple but ugly solution would be storing all your mprime files
(INI-files, p/q files, etc.) on an ext2 partition, and copy them
to/from the vfat partition automatically on boot and reboot. (How
you do this depends a bit on your distribution.) Of course, if you
don't use Windows very much, you could drop `Windows support' totally.)
/* Steinar */
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Date: Sun, 27 Jun 1999 19:14:14 -0400 (EDT)
From: Lucas Wiman <[EMAIL PROTECTED]>
Subject: Mersenne: Mersenne FAQ
All,
Here is a potential FAQ of the mersenne mailing list. At the moment
it only deals with factoring, but this should change.
Any suggestions for what should be in the FAQ? Any mistakes? Any
Clarifications?
- -Lucas Wiman
Q: How is factoring done on mersenne numbers?
A: We check a potential factor n of Mp by seeing if 2^p (mod n)=1
Q: But doesn't this mean using very large numbers, and computing 2^p?
A: Not at all, there is a very simple algorithm for finding a^b (mod c),
and here's how it works:
we set res=1,
then multiply res by a^(2^n) mod c whenever the nth binary digit of b is 1
a^(2^n) mod c is relativly easy to calculate through repeated sqaring of
a mod c (note that for any o,p, and m, (o^p)==(o mod m)^p (mod m) where ==
is the "congruent to" symbol)
Q: That's still a lot of factors!!!
A: Well we can significantly reduce the number of factors through some
simple theorems which state that:
(1) any factor of 2^p-1 must be of the form 2*k*p+1 for k>0
(2) any factor of 2^p-1 must be of the form 8*n+1 or 8*n-1 for n>0
(3) the smallest factor (other than 1) of any number must be prime
Theorem (1) reduces the number of cases to 1/(2*p) of all numbers.
Theorem (2) reuduces those to 1/2 of those.
Theorem (3) can reduce the factors down to only prime numbers, but that takes
too much time.
It is generally better simply to sieve out potential factors with very small
factors.
Q: But how can you sieve out small factors from potential factors?
A: Well, I think an example would be most illistrative.
Say we want to seive out all potential factors divisable by 3, 5, or 7.
Well we can create an array of 3*5*7=105 elements, with all elements set
to 1 (we'll call this array N). Then elementate all N[i] such that i is
divisable by 3, 5, or 7. Then we can keep a running value of 2*k*p+1 mod 105
then if N[2*k*p+1 mod 105]=0 we skip that factor, and go on to the next one.
Q: How much does all this actually help?
A: The sieving for small factors (up to 17), as well as theorem (2) can can
seive out about 32% of potential factors of the form (2*k*p+1).
Q: Wouldn't it be more effiecient to check prime numbers and see if they
divide as certain 2^p-1?
A: Yes and no. It is theoretically more effiecient to do this, however
this also produces uninteresting (and relativly useless) results for where p
itself is composite.
Q: A prime number cannot divide more than one mersenne number, so why not
create a table of all the prime numbers which divide mersenne numbers so
as not to duplicate work?
A: This isn't really workable because it would take longer to check the
table than it would to just check the factor.
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Date: Sun, 27 Jun 1999 17:16:02 -0700
From: "Robert Clark" <[EMAIL PROTECTED]>
Subject: RE: Mersenne: A few questions
> George has announced that the number that was suspected
> prime has been
> confirmed to be prime by different hardware and algorithms.
> Because of
> the EFF award money rules, we "GIMPS at large" will not be
> permitted to
> know what the exponent was, or who discovered it, until an
> article has been
> published in a peer-review academic research magazine or such.
>
> George has said he is working on getting such a publication,
> but that no
> publication date has been set. If he were to announce the
> prime NOW,
> prior to publication, however, it COULD void the
> discoverer's claim to the
> prize money, so I understand the caution in talking about it.
Well, the EFF has always struck me as a reasonable sort of
organization, not prone to arbitrariness. Have you asked them, George,
whether you can submit the exponent to them to establish precedence,
then announce it publicly so we can all know what it is, and then wait
for the money until after the article is published? I presume the
publication requirement is a way for them to be sure the claim has been
verified to be real.
Payment after publishing makes some sense, but why would not being able
to announce until after publishing make any sense if you've already
established you knew the exponent first?
Or have I completely missed some pertinent point?
Robert
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Date: Sun, 27 Jun 1999 18:10:35 -0700
From: Luke Welsh <[EMAIL PROTECTED]>
Subject: Mersenne: M38 in the news
http://www.oregonlive.com/news/99/06/st062601.html
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Date: Sun, 27 Jun 1999 21:08:15 -0400
From: George Woltman <[EMAIL PROTECTED]>
Subject: RE: Mersenne: A few questions
Hi all,
At 05:16 PM 6/27/99 -0700, Robert Clark wrote:
>Well, the EFF has always struck me as a reasonable sort of
>organization, not prone to arbitrariness.
To keep everyone up-to-date:
The Fibonacci Quarterly, an academic refereed journal, is willing
to carry an article. This should satisfy EFF's requirements.
Now that I know what journal we are publishing in, I submitted
the official claim with EFF today. Assuming they respond that the claim
is all in order, then we should be able to announce shortly thereafter.
The EFF can't pay until after publication, but EFF has never said that
GIMPS can't announce earlier. Fibonacci Quarterly has no problem with
us announcing before publication.
The only fly in the ointment is EFF's rule 4F "...submit a citation and
abstract of a published paper...". They could argue that my submitted claim
which references an upcoming issue of Fibonacci Quarterly is not a published
(past tense) paper and I should resubmit the claim once it is published.
Keep your fingers crossed,
George
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End of Mersenne Digest V1 #589
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