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At 10:50 AM 2/9/00 -0500, Jeff Woods wrote:
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.
Q: What does this have to do with
it thrashes, my 486 is essentially rendered useless.
The P-1 step would become a fourth work assignment (to be done after a
number has been trial factored, but before it has had an LL test run on it).
Much like how the trial factoring and LL test are done on separate computers.
-Lucas Wiman
/gmp (I think), there is a dos batch file), and compile
Will's program
(I think it is in the links section of the mailling list FAQ). But, under
windows, would George's program be the fastest (or are you using very large
exponents?)
Have fun,
Lucas Wiman
Hi there, I've experienced a problem. I'm wondering if anyone else has had
this problem:
GIMPS freezes my system after it runs for about 24 hours or so.
I have Windows NT 4.0 (SP5)
Dual Intel Celeron 433Mhz (BP6 motherboard)
128MB RAM
If I quit both GIMPS processes on my machine
- for an exponent around 35 million, it takes approximately
2000 CPU clocks i.e. 4 microseconds per test on a 500 MHz CPU.
This is a bit misleading, more accuratly, it runs in time proportional
to lg(p)*lg^2(n), where p is the exponent, and n is the potential factor.
Just call me "Mr. Pedantic".
e.g. 217=7*31
sqrt(217)~=14.737, but 3114.73.
-Lucas Wiman
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If you're factoring numbers in the 1165-1166 (bit) range, the first factor
could be anywhere in the root(1165) - root(1166) range i.e. 3413 - 3414 bits
long !!
No, in the x-y bit range (remember that n bit integers are about 2^n) the
first factor could be x/2 to y/2 bits long
why is f(0) in an ll test = 4
Well, it needn't be. There are a number of other possibilities, but 4 is the
smallest...
also when i went to sshool (i'm 34) 7 mod 8 == 7
i have seen 7 mod 8 == -1
is -1 == 7 mod 8
which is more correct
They are both correct. If you have a congruence
Now that double checking has surpassed exponent 4194304, what happens to all
those machines out there running the old v17 that produced incorrect results
for LL tests above that exponent? (I don't have any, but they almost certainly
exist.) Up to this point they will have been successfully
I may have missed this, but was there any final judgment on the Meganet
Corp. claim that they had a "deterministic and polynomial-time" prime
test? There were some discussions on the list early this year when they
first made their claim. But I don't recall reading about any resolution.
Well, I'm prepared to have a go. Could we tighten up the spec a bit?
(a) There's also been some interest in something else that Prime95
doesn't do - trial factoring 2^p+1.
(Note that this form also form also has factors of the form k*p+1.)
(b) I assume we're only interested in 2kp+1
To The MBase:
Haven't received any forwards since Friday Oct. 29th...
I believe the problem lies at the source. I don't think that anybody
has been sending mail to [EMAIL PROTECTED]
-Lucas
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But while watching the movie Pi, it occurred to me that since the number
Pi has *nothing* to do with base 10, there would be no repitition in the
digits in any approximation of it in base 10 (or any other integer base,
for that matter). The sequence of digits *should* be completely
Noone could call their full decimal
name in a life time, allthough it is finite in length.
If I remember correctly, someone once recited millions of digits of Pi from
memory in three days. Heh.
You probably don't :). I believe the world record was around 40,000 digits.
I find it highly
I was pretty surprised that the extrapolations for M38-M42 (all that I
did) were *exactly* the same for both methods of extrapolations,
Your two methods are equivalent.
I know, but I thought the algorhythms for fitting linear exponential
lines would differ enough to result in at
According to the "Where is the next larger Mersenne prime?" page --
http://www.utm.edu/research/primes/notes/faq/NextMersenne.html the
Wagstaff conjecture suggests a slope of 3/2, which I believe wouldn't look
so bad.
No. Wagstaff's conjecture predicts a slope of around 1.47,
STL: what was the line you fit to the log log of the known mersenne's
exponents, and what was the error (r^2) ?
I'm assuming r^2 is like, the amount of error for a given line fitting a
set of data.
When I fit an exponential line to the 1st 37 mersenne prime's exponents
(since I believe
Conjecture time: The prime number earning the $150K prize will not be a
Mersenne.
This isn't so much a conjecture as a prediction (there is a difference).
A conjecture is a very specialized prediction.
Why do I say that? Even with processor speeds increasing, we have a good
idea how long
Suppose M(x) is the number of primes p = x for which 2^p - 1 is prime
Lenstra, Pomerance, and Wagstaff all believe this [an early conjecture by
Gillies] and in fact suggest that ?? M(x) ~ e^gamma log x ?? where the log
is to base 2.
Hence, my new conjecture:
?? M(x) ~ e^gamma
Personally, I think the big problem with regards to this is not people
quitting so much as the possibility of major hard-drive failure etc. on the
testers. I doubt many of them keep good backups
NO EXCUSE! A Zip drive or a CD-R is inexpensive and effective; one
will service many
.TXT files are the one true document file. Pure ASCII bring it on!
Mathematical notation in .TXT isn't easy, though. It's a conspiracy.
It works well enough. Most of it is pretty standard, and TeX fills in
the rest
$$\int_{0}^{1}\sqrt{1-x^2}dx={\pi \over 4}$$
Yeah!
FOR GOD SAKE GIVE ME
What's the best way of finding the number of decimal digits for the number
2^p-1 ?
Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
^^^ ^^
Q5.3. It's no good unless people read it!
-Lucas
Hmm... I just changed my worktodo.ini to Test=5014947,63 (where's the 63
come from ? it was used for the last number I was assigned).
It's saying "Error: Work-to-do file contained composite exponent: 5014947"
I suppose that means it's already been tested found to be non-prime ?
There is now a prize for factoring Fermat numbers too.
Neat. Where's the info ?
I think Richard Crandall is offering a prize for Fermat factors
(http://www.perfsci.com). John Selfridge is also anouncing a prize
for factors of various numbers which "ought to be prime". I don't
think that
I think trial factoring is done to 2^68 for an exponent around 33 million.
Thus your chance is 2 * 68 / 3300.
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
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.
Anyone who has data which they wish to make publicly available may
wish to take advantage of my FTP server.
Downloads from the server are already available by anonymous FTP
(ftp://lettuce.edsc.ulst.ac.uk/gimps), I keep copies of many Mersenne-
related programs and also George's database
I would like to look for a factor of a mersenne prime in a specific area.
For example, for a mersenne exponent of say 40,000,000. I want to use the
Prime95b program, (v19 I guess), to search for a factor in a specific range
from, say, 2^40 to 2^50. I do not understand the ECM factoring
When this is cleared up, it will make a good FAQ.
Who maintains the FAQ list? Do you agree the answer here is a good FAQ?
I would like to look for a factor of a mersenne prime in a specific area.
For example, for a mersenne exponent of say 40,000,000. I want to use the
Prime95b
I must admit, I dallied (albeit very briefly) with the bovine rc5 client.
Gack! About the only manufactured challenge comes from ID software.
Should read About the only more manufactured challenge comes form ID
software. :)
-Lucas
I've been wondering about this lately...
If we are to begin P-1 testing on larger exponents, this implies
lower trial factoring limits (though possibly only by 1 or 2 bits).
Now, P-1 requires an investment of a GCD on two numbers each of
similar size to Mn. So, since we are investing this much
All (and especially Chris),
Yesterday (and the day before), I went to the Illinois number theory
conference.
There (2nd talk of yesterday) J. P. Selfridge announced that he would
give away $1000 US for any factor found of a number which ought to be
prime (he provided a list). On that
All (and especially Chris),
Yesterday (and the day before), I went to the Illinois number theory conference.
There (2nd talk of yesterday) J. P. Selfridge announced that he would
give away $1000 US for any factor found of a number which ought to be
prime (he provided a list). On that list was
Oopsy. That should have read J. L. Selfridge
-lucas
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Of course, the sequence that still remains unknown is
2
M(2)=3
M(3)=7
M(7)=127
M(127)=170141183460469231731687303715884105727
M(170141183460469231731687303715884105727)=???
Yes, this sequence is interesting, but if someone finds a way to prove/
disprove the primality of M(M(127)), I
Might be nice to display the percentage out to an accuracy that changes
every hundred iterations. Hmm, looks like that's an integer of the
percentage, not rounded. Guess it doesn't matter. For the one I'm
working on it looks like 3 decimal places would be needed to see a change
every 100
But as everyone on this list knows, any factor of a Mersenne
number looks like 2*k*p+1 for N=2^p-1. Plugging this into
the above equation gives
q=2*k*p+1
q-1=2*k*p
Correct.
Doesn't this mean the lower bound on a "p-1" method search should
be greater that the Mersenne exponent
Ackk! That was rather inexact wording. Let me try again...my
Celeron 400 based systems crunch exponents in the 384K FFT range at about
the same speed as George's PII-400 machine. However, at the 448K FFT size,
George's machine appears to be 20% or more faster than my Celeron 400s.
One version of Linux has paid the bill and passed the test, so at least one
version of Linux is Unix.
If you wanted to be picky, you could always say that a version of _GNU_/Linux
has passed the test... The tests aren't for the kernel only, are they?
But "GNU" stands for "GNU's Not
By the end of next week I will remove the mailing list
archive search function. The index for the archive occupies
1/3 of my disk quota. Considering how little it has been
used, I can make better use of this disk space. I hope
someone else would be willing to rebuild the index and host
revealed 2^60). You should
update primenet's files pertaining to this off of those at
http://www.informatik.tu-muenchen.de/~kruppa/M33M/index.html
-Lucas Wiman
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Mersenne
Now my question to the list is, would I be better off having these machines
participate in GIMPS via MacGIMPS on MacOS or should I run mprime on Linux? My
main concern is performance. Which performs better on the same hardware,
MacGIMPS or mprime? I would guesstimate that they all have ~32 MB
If I understand P-1 factoring correctly, then using it to a stage one
bound of k to try to factor M(p) will find all possible factors less
than or equal to 2*k*p + 1.
Yes, of the form n*p+1 (not 2*n*p+1 :). This is for the simple reason
that every power of a prime =k must divide Q (due to
Numbers above are factored to
- ---
71002^72
57022^71
44152^70
35102^69
28132^68
21592^67
17852^66
13382^65
825 2^64
Isn't this the "optimal"
I've just finished trying to factor F24 using the P-1 method
with a bound #1 of 1,000,000. No factor was found.
Unfortunately, I do not have enough memory to run stage 2.
So, is there a volunteer that would be willing to run stage 2 on
their computer. It will take about 3
2. Is overclocking of my Celeron 333 a good idea? I probably can't do
it just now (EPoX P2-112A motherboard isn't made for such purposes),
but I could upgrade it in about 2-3 months' time.
No. This can often lead to errors, and newer CPU's are on the edge of
sanity when it comes to heat
The last time I did timings like this - admittedly, probably over a
year ago, but the mers package hasn't changed much since, especially
in terms of performance - this is wrong. SPARC LL testing -
especially with MacLucasUNIX - is much closer to matching Prime95 LL
testing than SPARC trial
If B's checkpoint info doesn't match A's, the exponent is issued to
another system C which calculates to the first checkpoint check in,
as before. If we now have two matching residuals, the two systems
supplying them are told to continue and the system with the "wrong"
result is told to
rate should increase with the square of the
exponent (plus change). This means that if 1% have errors at 7mil, 22% will
have errors at 30mil.
-Lucas Wiman
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That is to say when
one computer finishes to X%, it reports its 64-bit residue to primenet, and
waits for the second computer working on the same LL test to do the same.
Until the other (slower) computer reports in, the (faster) computer works on
another exponent.
Not at all. The
Having one power supply per computer, besides the convenience of having it
self contained, is the fact that the fan in a power supply is used not only
to cool the supply itself, but to circulate air throughout the entire case.
This is especially true for ATX spec cases and supplies, but is a
They will have only a floppy attached, which will run once (at boot up).
Are these basically just "NC's", with a boot floppy and a network Windows 95
location?
You lousy Microsoft freaks! They will be linux boot disks that will run
either Mprime v19, or Brian's factor95.c. We haven't decided
If Pepin's primality test was run on F24, we could determine if F24 was
composite or prime. The chances that F24 is a psuedo-prime (a composite
pretending to be a prime) are very slim so it is worth the time to run the
test. I estimate the time required would be 6-9 months and we could do
Can we cut the crap on this list?
Hear, Hear! Especially the neo-poaching thread. This is simply too
volatile and pointless a thread to waste our time on.
Quite frankly, I'm also getting kind of annoyed with "When do you think
we will see a [power of ten] digit prime?" type questions also,
1) What is the approximate P-90 computing time to test for primality
for a 1, 10, 100 million ( 1 trillion!) digit Mersenne Primes?
Generally, a billion comes after 100 million, but I can't remember how the
British and other countries work with billions. And I don't think that many
people run
OK - I'm not a math guru, and am new to the list. I can't find a
recent archive so I hope this isn't a repeated question. Perhaps
there is someone out there that could help me get some answers to the
following basic questions ideas. Feel free to send me email
directly. Thanks!
Check
Unless George/Scott set some legal mumbo jumbo that ties into use of the
program/source/services, they're simply not "entitled" to any prize money.
I'm forced to agree with Aaron, aparently at gunpoint :-) (and I said this a
while ago, BTW). Even if they (George and Scott) did this, then there
At present we find approx. 1% of the LL
test results submitted are incorrect.
That figure seems a tad high. After double-checking, there would be a 0.01%
chance that BOTH tests had failed, which seems very high to me.
Well, the likelyhood that a failure occurs may be 1%, but the likelyhood
that
All,
On the subject of a hardware type FAQ/section in FAQ, here are the few
FAQs that I could think of off the top of my head. I'm guessing others
can think of more.
I'm really not qualified at all to write about win95/nt/os2 questions.
I've never used os2/nt, and I haven't used win95 in over 3
Sorry if this is sending out twice, I wasn't sure that it got sent the
first time...
As I understand the Pollard P-1 algorithm, the time required for 1 bit of
the exponent would be equal to 1 iteration of the LL test (one squaring
mod Mp), so values wouldn't be that hard to create.
(timing
I added the FAQ to the following search engines:
Lycos
Excite
Yahoo
Altavista
AOL netfind
Webcrawler
HotBot
InfoSeek
-Lucas
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privately.
Or we could have monthly meetings in coffee shops like 2600 does...
That'd be keen.
-Lucas Wiman
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there would be many people who would be willing to do this,
but I don't think it would make us look good to write and ask to give
it away.
-Lucas Wiman
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Mersenne Prime FA
se
people, so if any of you have problems with that, tell me.
I will register the FAQ with the major search engines in the next few
days. I really hope this FAQ reduces repetition on the mailing list.
thank you all,
Lucas Wiman
U
All,
I tried to add an FFT section to the FAQ, and here is what I got:
(please add, or correct errors...)
-Lucas Wiman
P.S. Sorry if the FAQ is causing more traffic than it is supposed to prevent
;)
===
Q6.1: How can we
Optimizing the -2 step down to nothing (with the amazingly high estimate of
1/50,000th of a sec) would gain only .3 CPU years for all exponents in the
7million range. I was just making sure.
-Lucas Wiman
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Does anyone know where I can obtain a copy of ECPP for the intel
platform, specifcally Linux?
I apologize for the slightly off topic-ness of this question...
thank you,
Lucas Wiman
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All,
I recieved a message pointing out a possible error in my FAQ:
Speaking of Q2.6, I've heard that with Crandall's DWT, the subtraction
2 step costs nothing at all. It's done automatically within the
transformation. Try checking this with George Woltman.
Is this true?
-Lucas
Could someone show me a proof for(if there is one):
The number 2^p-1 has factors in the form of 2pk+1
where k is any positive integer.
try http://www.utm.edu/research/primes/notes/proofs/MerDiv.html
-Lucas
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atively independent
of the set of numbers.
Note that in the set of mersenne prime exponents (so far), the leading
digit 1 (in decimal), turns up 10 times as opposed to the 4.2 times
expected by equal leading digit distribution...
-Lucas Wiman
P.S. I seem to recall this being mentioned (benford
*log(2)) time.
Hope that helps,
-Lucas Wiman
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All,
In the book _Primes and Programming_ Head's method of multiplying
two numbers mod n is mentioned. Is this actually more effiecient
than simply multiplying the two numbers and taking the modulus?
If so, is it implemented in the various mersenne factoring programs
in use?
Thankyou,
Lucas
to mention the prize money involved in the finding
of the 38th. This should speed us on our way to victory...
-Lucas Wiman
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Actually I think mentioning the prize money might constitute
deception, since the next prize isn't in reach, yet. However, I agree
with your sentiment, except - _what_ "victory"?
It might be interpreted as a deception, I suppose...
"victory" would of course be finding anot
wn section, or in the beginning stuff section. I'll probably add this
question sometime tomorrow.
-Lucas Wiman
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)
If you still have questions after reading the relevant sections of the
FAQ, by all means send them to the list!!
Thank you,
Lucas Wiman
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whether or not there are
an infinity of Sophie Germain primes of the form 4*n+3. I imagine there
would be, as there are an infinity of primes in the form 4*n+1 and 4*n+3.
-Lucas Wiman
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Therefore the geometric mean must be between a and b.
I think that you mean that the geometric mean of two successive mersenne
numbers is 2 raised to the (1/e^gamma) raised to the index of the mersenne
numbers.
-Lucas Wiman
Unsubsc
The page belongs to a previous record holder...
Took me about 30 seconds to find it.
It's nice to see a thirty-eighth line in /root/math/ref/mers...
-Lucas Wiman
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aren't searching for the right people. He didn't say it was a *GIMPS*
record holder. You've just gotta know where to look.
Help! :-)
All right, here's a hint: he held the record for largest prime, which was also
a non-mersenne prime. Check the largest prime by year...
-Lucas Wiman
place.
-Lucas Wiman
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of factors so non-linear?
-Lucas Wiman
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math to do it. I also like the idea of a history
section. I have read something about the history of GIMPS, as well as some of
the list's archives, but I have only been here since February...
-Lucas Wiman
Q: What are mersenne numbers?
A: Mersenne numbers are numbers of the form 2^p-1, (2
from some weeks back to get the exact first exponent.
The formula for determining the number of digits in Mp is p*log_10(2).
Therefore, to find the first one with 10^7 digits, we find ceil(10^7/log_10(2))
which is 33219281.
Yes, this is going in the FAQ...
-Lucas Wiman
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
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
, (that should take a
while). What do you mean by stragglers?
-Lucas Wiman
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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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Did I write polynomial anywhere ? I said deterministic primality proving algorithm.
I thought that perhaps you mispoke, since your statement seemed incorrect.
My point was that trial division is deterministic, but you said there was
only one such algorithm.
-Lucas
"Polynomial?"
I think and please correct if I am wrong that trial factorisation
using long division requires O(sqrt(n)*log(n)) operations.
sqrt(n)*log(n) is polynomial in n e.g. it is less than n^2.
Presumably when measuring the order of factorisation
algorithms you guys use n ~ e^x and
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