Re: Mersenne: Re: mprime startup at boot-time
On Tue, Oct 26, 1999 at 09:19:24PM -0400, Darxus wrote: If such a person can gain physical access to your machine, he owns it already. There is no defense against physical access. There's a BIG difference here: 1. Unlimited physical access. 2. Physical access for three seconds or so... And if that *were* useful information, it'd be almost as easy to find it in /var/log. If that was public. I'm afraid I just don't agree with you :) As I said, it it paranoia ;-) /* Steinar */ _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: More Schlag
hi,
recently Joth Tupper posted a message with the following
"I seem to recall a non-intuititive theorem about rational approximations to
numbers (this is from c. 1968). If you can approximate a number too closely,
then it is transcendental. S.Lang wrote a book on trancendental numbers and
degrees around 1973 and a precise statement might be there.
Does anyone recall this?"
Yes!
This is the result that Liouville proved to show that Liouville's number
must be transcendental. The result is known, funnily enough as Liouville's
theorem.
It says that if \alpha is an algebraic number of degree n \geq 2, then there
exists a constant c(\alpha)0 such that
| \alpha - p/q | c(\alpha)/|q|^{n}
always holds.
Or if you prefer, for any mn,
| \alpha - p/q | |q|^{-m} call this (*)
has only finitely many solutions.
At the root of the proof is the extremely "profound" fact that there are no
integers between 0 and 1 (or rather that the integers are discrete):
Let f(x)=x^n+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} be the "minimal polynomial" for
\alpha over the rationals. Let p/q be a rational number.
Then
q^{n}f(p/q) is an integer, it cannot be zero because f(x) is the minimal
polynomial for \alpha, therefore
|q^{n}f(p/q)| \geq 1.
f(x) =(x-\alpha)g(x) where g(x) is some other polynomial, so
|q^{n}(p/q-\alpha)g(p/q)| \geq 1 or
|p/q-\alpha| \geq |1/(g(p/q)q^{n})|.
One can estimate g(p/q) quite easily and use this estimate to get the value
for c(\alpha) mentioned above.
You prove that the Liouville number (or any similar type number) is
transcendental from this result is by noting that the rational
approximations are getting more and more accurate at each step (you are
basically adding lots more zeros each time). But Liouville's theorem forbids
this, if the number is algebraic. Thus the number is transcendental.
It can be formulated somewhat different yet again (in terms of "heights" of
numbers) and it is an extremely important result that is used almost
everywhere in diophantine analysis and transcendental number theory (so
ultimately most proofs in these areas depend upon cleverly showing that some
"deep" result depends on the fact that there are no integers between 0 and 1
:-)).
Liouville proved his theorem around 1840. It has since been improved.
Thue (1909) showed that you can weaken the condition on m in (*) above to
mn/2
Siegel (1922) shows that m2\sqrt{n} suffices
Dyson and Gelfond independently showed in the 40's that m\sqrt{2n} suffices
and finally (sort of) Roth in 1954 showed that m2 suffices.
Roth won a Fields medal for this (and some other) work.
Does that answer your questions?
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Re: Mersenne: (2^p-1)/p == 0 (mod 3) ?
"Vincent J. Mooney Jr." wrote: 5 is an odd prime. 2^5 = 32 and minus one, is 31. 31 is not divisible by 3. At 07:50 PM 10/26/99 +0200, you wrote: Hello all, a simple Number Theory question. Is always (2^p-1) / p ,odd prime p, divisible by 3 ? Well, I managed to mess up the equation so badly that noone could possibly make any sense of it. It should have been: (2^(p-1) -1) /p, and by now I figured out why these numbers are divisible by 3 for odd (and not just for prime) p3 (if you kick out the /p, its for p2). a^k == 1 (mod a^k -1) a^n == a^(n-k) (mod a^k -1), n=k a^(nk) == 1 (mod a^k -1) (or a^(nk)-1 == 0 (mod a^k-1) ,which shows that Mersenne primes must have prime exponents) Since 2^2-1=3, 2^(2k)-1 == 0 (mod 3) for all integer k and, in the above formula, 2^(p-1)-1 == 0 (mod 3) for odd p. Thats what happens when you ask a question before thinking it to the end, sorry for the confusion. Ciao, Alex. _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Prime Theory
Hi all, I am quite new for the prime numbers subject, and when go deeper in theory of them I encounter a serious problem - all docs and faqs are in "mathematical english", but "mathematical english" is not my favorite one :). Please let me know, if there is somewhere in the "netland" such a documentation in pure mathematic form ( I mean math equations instead english poetry ) in TeX, pdf, ps or similar "graphic" format, which I can understand...( I hope :))) ). Thanks in advance/Remi. _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: More Schlag
Thanks! Good Stuff!
Joth
PS: There is no Nobel Prize in math. This is perhaps unfortunate, as the
winner of a Nobel Prize
seems to get over $500,000 (US) in cash and generally a new building at the
university of choice.
The closest math counterpart is the Fields Medal which does include a prize:
$15,000 Canadian
(or about $10,000 US) and sometimes a corner office.
Ah, the peaceful uses of dynamite.
- Original Message -
From: Alan Simpson [EMAIL PROTECTED]
To: [EMAIL PROTECTED]; [EMAIL PROTECTED]
Sent: Wednesday, October 27, 1999 2:07 AM
Subject: RE: Mersenne: More Schlag
hi,
recently Joth Tupper posted a message with the following
"I seem to recall a non-intuititive theorem about rational approximations
to
numbers (this is from c. 1968). If you can approximate a number too
closely,
then it is transcendental. S.Lang wrote a book on trancendental numbers
and
degrees around 1973 and a precise statement might be there.
Does anyone recall this?"
Yes!
This is the result that Liouville proved to show that Liouville's number
must be transcendental. The result is known, funnily enough as Liouville's
theorem.
It says that if \alpha is an algebraic number of degree n \geq 2, then
there
exists a constant c(\alpha)0 such that
| \alpha - p/q | c(\alpha)/|q|^{n}
always holds.
Or if you prefer, for any mn,
| \alpha - p/q | |q|^{-m} call this (*)
has only finitely many solutions.
At the root of the proof is the extremely "profound" fact that there are
no
integers between 0 and 1 (or rather that the integers are discrete):
Let f(x)=x^n+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} be the "minimal polynomial"
for
\alpha over the rationals. Let p/q be a rational number.
Then
q^{n}f(p/q) is an integer, it cannot be zero because f(x) is the minimal
polynomial for \alpha, therefore
|q^{n}f(p/q)| \geq 1.
f(x) =(x-\alpha)g(x) where g(x) is some other polynomial, so
|q^{n}(p/q-\alpha)g(p/q)| \geq 1 or
|p/q-\alpha| \geq |1/(g(p/q)q^{n})|.
One can estimate g(p/q) quite easily and use this estimate to get the
value
for c(\alpha) mentioned above.
You prove that the Liouville number (or any similar type number) is
transcendental from this result is by noting that the rational
approximations are getting more and more accurate at each step (you are
basically adding lots more zeros each time). But Liouville's theorem
forbids
this, if the number is algebraic. Thus the number is transcendental.
It can be formulated somewhat different yet again (in terms of "heights"
of
numbers) and it is an extremely important result that is used almost
everywhere in diophantine analysis and transcendental number theory (so
ultimately most proofs in these areas depend upon cleverly showing that
some
"deep" result depends on the fact that there are no integers between 0 and
1
:-)).
Liouville proved his theorem around 1840. It has since been improved.
Thue (1909) showed that you can weaken the condition on m in (*) above to
mn/2
Siegel (1922) shows that m2\sqrt{n} suffices
Dyson and Gelfond independently showed in the 40's that m\sqrt{2n}
suffices
and finally (sort of) Roth in 1954 showed that m2 suffices.
Roth won a Fields medal for this (and some other) work.
Does that answer your questions?
__
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Mersenne: Trial-factorers
I'm looking for program(s) capable of trial-factoring prime exponent Mersenne numbers (using 2kp+1) meeting the following requirements: 1) Capable of trial-factoring any exponent 1 (at least to some considerably large number, say 1 trillion?) As I recall, Brian [Beesley] mentioned something once about having a program that could test an exponent of an arbitrary size... Brian?? 2) Capable of testing a factor of any size. (even over the 2^76 limit of Prime95). I just know somebody is going to have to mention the time involved in testing factors of such a large size. Let me just say, I realize *exactly* how much time would be required... 3) Capable of trial-factoring a range of k's. (example: from k=1000 to k=2500) It would be best if all three of the requirements could be fulfilled by a single program... Can anybody be of some help??? Eric Hahn _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Questions about prime.ini syntax, and hardware advice.
-BEGIN PGP SIGNED MESSAGE- Hash: SHA1 Hi. I'm currently trying to configure the Time command, as listed in the docs, to get the prime95 client to function as follows. User ID=XYZABC Time=1-5/18:00-0:00,1-5/0:00-08:00,6-7/0:00-24:00 (reset of Prime.ini) Anyhow, it doesn't seem to work, and I tried a few other syntax forms, but what would usually happen is that either the app runs at 6:00PM and then goes to sleep at 6:01. I inserted the line Priority=1 after the Time line, and it doesn't do much. I couldn't find an explanation of the time variables on the FAQ, I'm hoping someone's been down this road before. I don't know if the undocumented commands are being supported, or it's been discussed already. I'm using the prime.ini=yourfilename and local.ini=yourfilename. Anyhow. If I try to start the Prime95 Program, with the renamed.ini it will give me the user information dialog box. And will prompt to download new exponents (after doing a self test). if I leave the prime.ini and local.ini files without changing. It seems to work, except for an odd inconsistency that might be due to me adding all of these commands to my .ini file LockUserInfo=1 (Unsure if this is an Undocumented as it appears on the FAQ now) PercentPrecision=4 prime.ini=newfilename local.ini=newfilename worktodo.ini=newfilename prime.log=newfileaname prime.spl=newfilename resultst.txt=newfilename workingdir=C:\winnt\blahblah\blah I know it's slightly off-topic, but I am in need of advice as far as how well Prime will run on this machine. I currently have an order in place for an Kryotech Cool Athlon™ 900. For anyone who hasn't seen it. http://www.kryotech.com/maintext900.asp Anyhow, it's an Athlon that's running at 900Mhz, besides the novelty value of that. How fast would something like this get through an average LL test? Something in the 10202624 range? I won't know until it it arrives, and this is somewhat assuming that Kryotech doesn't announce the Super G product until then, and/or they ship on schedule. Anyhow, any comments on the above are welcome. -BEGIN PGP SIGNATURE- Version: PGPfreeware 6.5.1 for non-commercial use http://www.pgp.com iQA/AwUBOBejG6svR6n7TjeiEQL7OwCfc3tPSrChhK6A8KHGjJ8To+nXFAwAnRBh E9arZvjP3zwwyoPtdb9XDUZz =sEYB -END PGP SIGNATURE- _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Questions about prime.ini syntax, and hardware advice.
Albert Garrido wrote: I'm currently trying to configure the Time command, as listed in the docs, to get the prime95 client to function as follows. User ID=XYZABC Time=1-5/18:00-0:00,1-5/0:00-08:00,6-7/0:00-24:00 (reset of Prime.ini) If you're trying to run from Midnight to 6AM and 6PM to Midnight on Mon-Fri, the time line should read as follows: Time=1-5/0:00-08:00,1-5/18:00-24:00,6-7/0:00-24:00 Note Midnight is 0:00 when starting the day and 24:00 when ending it. It's exactly like what you have for Sat-Sun. It's also best if you put them in order by time (you'll notice I switched the two Mon-Fri times around). There's not usually any problems, but glitches do happen. I don't know if the undocumented commands are being supported, or it's been discussed already. I'm using the prime.ini=yourfilename and local.ini=yourfilename. Perhaps somebody else can help you here. They look okay, but I have never used them myself (no need!) I know it's slightly off-topic, but I am in need of advice as far as how well Prime will run on this machine. I currently have an order in place for an Kryotech Cool Athlon™ 900. [...] How fast would something like this get through an average LL test? Something in the 10202624 range? [...] Based on the known information about LL-testing times, Prime95 should be able to LL-test a number such as 10,202,623 in 2 to 2.5 weeks. A 10M digit exponent could be tested in about 7-8 months _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
