The first 32bit intel processor was introduced around '88 -89'.
the 486. It wasn't really fully utilized for 5 years. (I know Unix
Not true. The iapx432 came out in 1983 or there about. It was not a
commercial success, being many years ahead of the technology. It had a
fascinating
I got the idea to do some factoring with my now slower-than-average
machine (a P133), but I don't want to factor at the current
assignments
(in the 9M range); instead I would like to fill up the factor
limits of
small exponents to some common value (56 bits or 58 bits or so).
If what you
Seconded!
If, on the other hand, you'd like to contribute to another project, please
consider joining ECMNET and, especially, the client server version described
at http://www.interlog.com/~tcharron/ecm.html
Paul
-Original Message-
From: Aaron Blosser [mailto:[EMAIL PROTECTED]]
Sent:
Here's what seems to be an interesting little buglet and/or incompatibility.
The workaround is simple and I hope that the information below will save
others from having to discover it for themselves.
My home box is on the web only very intermittently. It finished a LL test
several days ago and
The Gaussian primes are 1+i (and associates); integer primes
of the form
4n+3 (and their associates) and the factors of integral
primes of the form
4n+1. The latter can always be expressed as a^2+b^2 and so their
factorization into Gaussian primes is (a+bi)(a-bi).
The "smallest"
I didn't know they had anything automated. That's a big plus for them.
Yes, and very smoothly it runs as well. I'm hosting the master server. I
also run sub-servers (one a slave to the master, another on my network at
home which is entirely independent) which are ideal for division of labour
Because the large the number, the more factoring is worth doing before
changing over to the computationally expensive LL test.
Paul
-Original Message-
From: Cyril Flaig [mailto:[EMAIL PROTECTED]]
Sent: 17 April 1999 00:02
To: [EMAIL PROTECTED]
Subject: Mersenne: factor
Why factors
At Paul Zimmerman's ECM page,
http://www.loria.fr/~zimmerma/records/ecmnet.html
the optimal B1 value listed for finding 50-digit factors is
4300, but
George's ECM factoring page uses 4400 for the same
purpose. Is one of
them wrong, or is there a reason for the difference?
The function being minimized, namely
probability of finding a 50-digit factor on one curve
-
time per curve
is flat near its minimum. Implementation and platform differences
can obviously affect the denominator
The $1500 I offered before the EFF contest began has since
gone toward more PrimeNet hardware. Between some GIMPSers,
a new prize pool of $.11 has been
proposed (the radix is 10 :-) If you think it's worth a few
The radix is always 10. I guess you mean the radix is
Ground rules are critical, but how about
/.1
where "/" represents the APL-style monadic divide or
multiplicative inverse.
1/.1
takes two.
Indeed, but ".1" represents 1 / radix, so 1/.1 is just radix.
The whole point of my tongue-in-cheek posting was to indicate how an
implicit
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
From: Kris Garrett [mailto:[EMAIL PROTECTED]]
I've noticed that with any odd number you can make the formula x^2 -
y^2 = n where n = the odd number and x - y and x + y are
factors of n.
I was just wondering if one could use the graph of a hyperbola to see
only the possible integer values
From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
On 14 Jul 99, at 1:26, Paul Leyland wrote:
Fermat observed that if one could represent an integer as the difference
of
two squares, its factorization would be at hand; he then gave an
efficient
(for the 17th Century!) algorithm
From: Olivier Langlois [mailto:[EMAIL PROTECTED]]
I've played a little bit FFTW and I've remarked that its performance can
vary a lot depending on how good your compiler is at optimization.
For instance, compiled FFTW code is far from optimal with MSVC++ 6. This
compiler doesn't fully use
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.
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
First, the important disclaimer, necessary because of my posting address: I
do not in any way speak for Microsoft in what I write below, but only in a
personal capacity.
In the W2k betas that have been issued so far, there is a "CMD" command
which does pretty much what you'd expect.
bash has
Aargh!!
I've spent *weeks* trial factoring some of those, time which could have been
better spent in ECMNET.
Unhappy.
Ah well, it's going to have to wait. I've got other things to do. In the
meantime, I'll just stop the programs.
Paul
-Original Message-
From: Eric Hahn
I've been reading up on the P-1 method ... am I missing something?
I don't think so, except perhaps for on possibility...
For a Mersenne number 2^n-1 with n prime we know that all the factors
are of the form 2kn+1 for some integer k. So the factorization of p-1
_must_ include at least
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.
Infinite to me means never ending.
BJB wrote:
Yes - I think we need this database - with or without savefiles,
it's a waste of effort to inadvertently duplicate work done before.
Since P-1 is deterministic (like trial factoring, but unlike
Pollard's rho or ECM) you should get the same result every if you use
the same limits
Is a decimal 23-digit numbers 111 prime ?
Could you tell me the answer with proof?
Giving proof of primality in a text message? Hrm. Interesting.
Ok, here's a proof by converse-of-Fermat:
Let p be the number above. Then p-1 has the prime factorization:
2 * 5 *
Oops. Minor typo:
Where I wrote:
All we need to do is find a value "a" such that a^((p-1)/ q) is not
equivalent to 1 modulo p for all of the primes q which divide p - 1.
(Actually, this is stricter than necessary, as a different "a" could be
chosen for each q, but never mind.)
I claim
-Original Message-
From: Spike Jones [mailto:[EMAIL PROTECTED]]
On the other hand, the existence of the EFF prize is a useful
tool in convincing companies to run GIMPS, for most IT
managers are quick to remind you that this *is* a business,
and it is here to make money. (Then
linux is a Good Move ... ceratinly, in its default state, it's at
least as secure (when used as a firewall) as anything emanating from
a certain purveyor of operating systems based near Seattle. It's
cheaper, too!
Please note: Seattle is about 5000 miles from where I am, despite my
From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
What I mean is that 10^100 = 3 + (10^100 - 3) is one possible
breakdown. But I don't know offhand whether (10^100 - 3) is prime,
and proving that it is might take some considerable time. If that
However, proving it is composite is very
From: Willmore, David [mailto:[EMAIL PROTECTED]]
value of 90 days. For some reason, I'm not allowed to get more than about
a
month of factoring work. It's starting to cut into my efficiency. Is
there
a way around this?
Try joining the people at the bleeding edge. I've been factoring an
A nasty thought just struck me.
It's well known that factors of M(p) are of the form 2kp+1.
It's also well known that factors found by the P-1 algorithm are of the form
(product of powers of small primes) + 1 where "small" means less than an
arbitrarily chosen limit B1. I omit stage 2 from
Will gimps support the search for the 1,000,000,000 digit prime?
How many years would it take for a PC to factor such a prime?
Not long at all. If it is prime, its only factors are 1 and itself.
Paul
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From: Pa'l La'ng [mailto:[EMAIL PROTECTED]]
1/3 + 1/5 + 1/7 + 1/9 + 1/15 + 1/21 + 1/27 + 1/35 + 1/63 +
1/105 + 1/135 = 1,
but you can't get an odd perfect number out of it.
Among these denominators you find no squares, so I really can't get a
perfect
number from this sum.
Really?
Can anyone explain this inconsistency in what the server believes the MSRC
account has done?
The individual report page gives back (at the head of a lot of other stuff)
Account ID LL P90* Exponents Fact.P90 Exponents P90 CPU
CPU yrs LL Tested CPU yrs* w/ Factor
From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
If P-1 does find a factor which is compound, then running P-1 again
with smaller limits will eventually recover a smaller factor. These
extra runs will obviously take less time than the original
Indeed, and with care one can usually choose
From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
On 14 Jun 00, at 4:49, Paul Leyland wrote:
It's a pity that a similar procedure isn't known for ECM,
or at least not
known to me.
Isn't the point that ECM finds numerically small factors much more
easily than it finds numerically
I wrote:
From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
If P-1 does find a factor which is compound, then running P-1 again
with smaller limits will eventually recover a smaller factor. These
extra runs will obviously take less time than the original
Indeed, and with care one
of finding two or more *on
the same curve* are somewhere between nil and negligible.
Paul Leyland said:
As long as the coefficients of the curve and the starting point are
recorded, we can re-run exactly the same computation, with the small
primes
curtailed as in the p-1 case, on the same curve
No, it decreases, and eventually becomes identically zero.
All prime factors of M(p) must be of the form 2kp+1. Once p reaches 2^58 we
can guarantee that there are no factors of this form which are less than 59
bits!
Paul
-Original Message-
From: Stefan Struiker [mailto:[EMAIL
I've read on the list some time ago that ECM takes, like
Pollard-Rho or
P-1, O(sqrt(f)) operations mod N to find a factor f. But
looking at the
factors found so far I find that hard to believe; according to that
And quite right too. It's just plain wrong. ECM runs in sub-exponential
Several members of the GIMPS team responded to my request earlier this year
to attempt ECM factoring of the number P773. Several thousand curves with
B1=11M and B1=44M failed to find a factor. The reason for this failure can
be seen in the announcement below; the earlier work gave a 50% chance
We could let prime95 decide the next election grin.
Give everybody a different prime number. Multiply the primes for
candidate A together, likewise for B.
And like in this election, unless we can completely factor
the results,
we wouldn't know who won, either.
:)
I note the
The problem is for the government to factor the huge composite number
for each candidate afterwards. Can someone give me a rough
estimate of
how long it would take a good supercomputer to check an arbitrary
100,000,000 digit number for several factors that are each a known 150
digit
Is it just me, or have we not had a database sync for a long
time between George and primenet? my personal results file
on primenet is showing values going back to March...
Nope, it's everyone. Mine go back to
6827629 63 0x7B026BBFAEE5C7__01-Feb-00 16:00 ROUGE
How
It is relevant here, and you need not worry about repeating other work.
The ECM implementation chooses an elliptic curve randomly each time it
is run --- at least, that's what it's supposed to do. If the random
number generator is any good at all, you have an entirely negligible
chance of
I wrote:
From: Simon Rubinstein-Salzedo [mailto:[EMAIL PROTECTED]]
Last night I was reading a number theory textbook, and I
thought of this odd conjecture: is one more than the
primorial of a Mersenne prime always a prime? I wouldn't
have a clue where to start on proving it, so if
From: Simon Rubinstein-Salzedo [mailto:[EMAIL PROTECTED]]
Last night I was reading a number theory textbook, and I
thought of this odd conjecture: is one more than the
primorial of a Mersenne prime always a prime? I wouldn't
have a clue where to start on proving it, so if anyone
had any
The user interface is quite desirable, but if no solution can
be found,
we have to use the nt version instead.
I've been using the NT version, under NT, for years. Whenever I want to
do something interactive with the computation I fire up Prime95. The
latter reads the same files that the
Speaking only for myself.
My take is that due to the long test times the instant gratification
that used to come fairly quickly even on a most speed machine is no
longer there. This will turn off a lot of casual testers.
This is certainly one reason why my contribution (the MSRC team) has
Well, I suppose RISC is about as close as you can get to
that. Microcode, for reasons best known to someone
familiar with CPU design, is not something
you can just reprogram on the fly...
I beg to differ. As someone who has written megabytes of microcode,
including an entire IEEE
Maybe we're not understanding what is meant by microcode...
The only CPU I've designed was a 4-bit system that didn't use
microcode to get it's work done (it was for a class), so I
can't claim direct experience, but I at least thought I knew
what the word microcode implied... a level of
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]]
(Aside - we're rather more likely to find a 75-bit factor than a 75-
digit factor. In fact, finding a 75-digit prime factor of a
no factors
known Mersenne number with an exponent under 80 million would
be a significant achievement
Is this a bug in the reporting software? I don't have the
tools to work it out exactly, but a 103-bit number should be slightly
larger
than 2^103, or
Nope. A 103-bit number N should lie in the range 2^102 = N 2^103.
Something really odd is going on.
Perhaps this small example will
Title: Message
Sorry
Ernst, but you're being misled by the success of the SPECIAL Number Field
Seive.
Indeed, we
can factor some fairly hard integers with over 200 digits, such as M727, R211
and (in progress right now) M751 but we can factor some easy integers with
millions of digits via
hmmm; my C8 easily resolves M4. Actually, M13 and M22 are
much nicer :-)
Agreed, though both 13367 * 164511353 and 7 * 73 * 151 * 631 * 23311 are
particularly nice at this time of year.
Paul
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Aargh, a typo 8-(
Agreed, though both 13367 * 164511353 and 7 * 73 * 151 * 631
* 23311 are
particularly nice at this time of year.
Should read 7 * 31 * 73 * 151 * 631 * 23311.
M5 got lost.
Paul
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From: Alexander Kruppa [mailto:[EMAIL PROTECTED]]
Torben Schlüntz wrote:
...
Besides I have the question: why does the advanced facor
algortithm of
prime95 somtimes find 2 factors? This happens eg. at M1289, has
108817410937 and 15856636079 as factors?
I'm not quite sure which
I think I know the answer to this... but want to
double-check to be sure...
While doing factoring... using ECM... factors up to:
15 digits is the equivalent of ~2^50...
20 digits is the equivalent of ~2^67...
25 digits is the equivalent of ~2^83...
30 digits is the equivalent of
If trial-factoring has been done up to 2^68... is it
possible to skip testing ECM curves for factors up to 15
and/or 20 digits... and go straight to testing ECM curves
for digits up to 25 digits???
Personally, I would go straight in at the 25+ digits level.
OTOH, if trial
From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
...
To get a real feel for this value for money question,
surely you have to factor in the system depreciation
cost i.e. the difference between the purchase and
residual values plus the total cost of the power
consumed over the working
George,
I think I've found two bugs in Prime95 or, at least, serious
misfeatures. I don't know whether they've been fixed in more recent
releases but as I'm using the program in a rather creative manner I
suspect not. The Mersenne list is Cc:ed so that if anyone else wishes
to use your program
Hey, please ascribe criticism where it's due. The article was written by a ZDNet
journalist, as clearly stated on the MSN page. ZDNet is a subsidiary of CNET
Networks, which is based in San Francisco. Same country, same coast, a lot further
south.
The comment by a computer scientist part
Why even bother with that? Just use gigaflops or something that is not
hardware dependent at all...
Ah, but which gigaflops?
Anyone else here old enough to remember Meaningless Indicators of Processor Speeds?
All gigaflops are not created equal, unfortunately. Wordlength alone can make a
I've found a few composite factors whilst running P-1 on
small exponents.
Working on M(32340) with P-1 turned out factors with several hundred digits, even when
the B1 limit was set very low. This experiment turned up a bug in Prime95 which
George has now fixed. The numbers were so
The reason that this answe ris somewhat of a lie is that the
prime numbers
used in cryptography are usually NOT the largest prime
numbers in the
world at the time, nor too close to it. (It'd be easy to
crack such keys
if they were limited to the 1000 largest primes -- then
you're
From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
usable by a single process is limited to 2 GBytes. (There is
a big memory
variant of the linux kernel which expands this to 3 GBytes,
but the point still stands).
FWIW, WinNT and its descendents can be booted with /3gb in boot.ini,
From: Brian J. Beesley [mailto:[EMAIL PROTECTED]]
I'm also working, at low priority again with a couple of
helpers, at completing triple-checking for all small
exponents (under 1 million). Why? So far as I'm concerned,
it's something useful for a couple of slow systems to
do whilst
Will Edgington and I try to keep our tables of Mersenne factorizations up to date by
synchronizng every week or so. The latest indicated that another Mersenne number had
been completely factored. Neither Will nor I have the resources at the moment to
prove the 2193-digit cofactor of M(7417)
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