Mersenne Digest Monday, 18 May 1998 Volume 01 : Number 363 ---------------------------------------------------------------------- From: gordoni@base.com (Gordon Irlam) Date: Sun, 17 May 1998 01:30:12 -0700 (PDT) Subject: Mersenne: List demographics Below is a breakdown of subscribers to this list based on the last components of their domain names: 364 com 120 net 116 edu 47 de 37 uk 28 ca 27 nl 20 fi 16 se 15 org 14 fr 12 au 11 jp 11 be 9 it 7 pl 7 no 7 es 7 at 6 za 6 dk 5 us 5 mil 4 sg 4 ru 4 gov 3 pt 3 cz 3 ch 2 si 2 nz 2 il 2 hk 2 ee 1 lv 1 lu 1 lb 1 kr 1 ie 1 hu 1 br 1 bm 1 bh 1 bg 1 ar Make of this what you will. In total their are 921 unique subscribers. 488 subscribers on the regular list, and 451 on the digest version of the list. gordon ------------------------------ From: Harry Kuhman Date: Sun, 17 May 1998 04:07:30 -0400 Subject: RE: Mersenne: Cryptography George Woltman wrote: >Richard Crandall worked out a method for using Mersenne Primes in >encryption. This was done while he worked at NeXT. They patented the >algorithm and it is NOT publicly available. ....... Grorge, How could it be patented and NOT publicly available? Patents are a matter of public record and must disclose the claims they are protecting, that's the whole point of them! And how could one avoid infringing on a patented algorithm if it was kept secret? If they patented an algorithm, can you provide the patent number? Harry ------------------------------ From: GivenRandy Date: Sun, 17 May 1998 06:04:49 EDT Subject: Re: Mersenne: Cryptography > How could it be patented and NOT publicly available? Patents are a > matter of public record and must disclose the claims they are > protecting, that's the whole point of them! And how could one avoid > infringing on a patented algorithm if it was kept secret? Very easy. You can often make the patent papers detailed just enough to distinguish from another patent, yet cryptic enough to hide the important details. That's what we did on a patent we got about 10 years ago. Randy Given GivenRandy@aol.com http://members.aol.com/GivenRandy public key at http://members.aol.com/GivenRandy/pgpkey.asc ------------------------------ From: "Gilles Polus" Date: Sun, 17 May 1998 11:43:38 +0200 Subject: Re: Mersenne: Cryptography Jean-Luc Cooke wrote: >> Gilles Polus wrote: >> >> I would know if there is any relation between Mersenne primes and >> cryptography. >> Sorry, but I don't speak english fluently ( I'm french). > >Yes there is. They don't use Mersenne in Cryptography becase they are >too easily recognized. For more info on RSA type cryptography check out >the following URLS: > >http://www.engsoc.carleton.ca/~jlcooke/Numbers/ >http://www.engsoc.carleton.ca/~jlcooke/RSA/ >http://www.engsoc.carleton.ca/~jlcooke/JLCauth/ > >If you have any questions or suggestions, email me and I'll see what >can do. I'm still new to this but very interested. Crypto is how I >tripped upon the GIMPS project. > >TTYL > >JLC I thanks you to answer me. I would know if the Lucas-Lehmer Test is the alone test to prove a number is a Mersenne Prime. ------------------------------ From: Paul Leyland Date: Sun, 17 May 1998 06:31:13 -0700 Subject: RE: Mersenne: Use Fermat First? > Q: Wouldn't it be faster to first test a number for pseudo primality > before going to the LL test? LL's take 50% more time than a > Fermat test > and Fermat tests eliminate a BUNCH of candidates (i.e.. all but > Carmichel's). Just a 2-PRP test should do. After that's Unfortunately, neither claim is true. All Mersenne candidates are strong base-2 pseudoprimes, and a strong pseudoprimality test to a base other than 2 *will* correctly characterize Carmichael numbers as composite with high probability. Paul ------------------------------ From: STL137 Date: Sun, 17 May 1998 16:02:39 EDT Subject: Mersenne: GRRR Gibberish STOP IT, ALL OF YOU! You don't realize how much time I, and everyone else, wastes looking for valuable info on math, while looking at USELESS gibberish like this: "begin: vcard > > fn: Jean-Luc Cooke > > n: Cooke;Jean-Luc > > org: Carleton University > > adr: 270A Dalehurst Dr.;;;Nepean;ON;K2G 4M8;Canada > > email;internet: jlcooke@ottawa.com > > title: Electrical Engineering Student > > tel;work: na > > tel;fax: na > > tel;home: na > > note: My web page address is: www.engsoc.carleton.ca/~jlcooke/ > > x-mozilla-cpt: ;0 > > x-mozilla-html: TRUE > > version: 2.1 > > end: vcard > > > ****************************************************************************** ** > > Vincent J. Mooney Jr. vincentj@erols.com - - -- Jean-Luc Cooke Carleton University, Electrical Engineering There is a 90% chance that right at this moment I am doing Calculus. God damn Newton, damn him to hell! - - ---------------------------------------------------------------------- Hyperactivity Unbounded http://www.engsoc.carleton.ca/~jlcooke - - ---------------------------------------------------------------------- AOL-IM ID: JLinOTTAWA http://register.netscape.oscar.aol.com - - ---------------------------------------------------------------------- ICQ# 4474419 http://www.mirabilis.com - - ---------------------------------------------------------------------- Email addresses jlcooke@ottawa.com jlcooke@magma.ca jlcooke@magmacom.com jlcooke@engsoc.carleton.ca jlcooke@chat.carleton.ca - - -------------------------------------" And the stuff about $500 per unsolicited commericial, etc.... Just turn the damn things off! Please! WE KNOW WHO YOU ARE! If I wanted to know your AOL-IM or your ICQ, I'd ask you PRIVATELY. Pleaseeeee. (By the way, this is not directed at whoever I cut and pasted this from. It applies to EVERYONE who does this. A short closer like I do is just finee.) STL137 ------------------------------ From: Will Edgington Date: Sun, 17 May 1998 13:43:15 -0700 Subject: Re: Mersenne: Use Fermat First? Chris Nash writes: I think there's a conjecture out there that if a Mersenne number of prime exponent (and some other restriction) is pseudoprime to base 3, then it is prime. Only a conjecture... This is related to - but not the same as - 'Conjecture B' of my mersenne.html page. Peter-Lawrence Montgomery found a counter-example for that, also in mersenne.html, most likely before I even thought of the conjecture. Since that counter-example shows that factors of prime exponent Mersenne numbers can be base-3 pseudo-primes, I suspect prime exponent Mersennes can themselves be 3-PRP. As for how often, I don't have any data, but writing a program to check using freeLIP would be straight-forward and much faster than Java. Will http://www.garlic.com/~wedgingt/mersenne.html ------------------------------ From: Chris Nash Date: Sun, 17 May 1998 17:16:36 -0400 Subject: Re: Mersenne: Use Fermat First? > I think there's a conjecture out there that if a Mersenne number of > prime exponent (and some other restriction) is pseudoprime to base > 3, then it is prime. Only a conjecture... > >This is related to - but not the same as - 'Conjecture B' of my >mersenne.html page. Peter-Lawrence Montgomery found a counter-example >for that, also in mersenne.html, most likely before I even thought of >the conjecture. Thanks for clarifying that Will. I remember mention of it in the mailing list before, but wasn't sure of the details. I guess if the Mersenne number itself is 3-PRP, that's a partuclar sub-case of the conjecture. >Since that counter-example shows that factors of prime exponent >Mersenne numbers can be base-3 pseudo-primes, I suspect prime exponent >Mersennes can themselves be 3-PRP. As for how often, I don't have any >data, but writing a program to check using freeLIP would be >straight-forward and much faster than Java. Very true. I'm thinking the quantity of prime-exponent 3-PRP Mersennes is small, but not small enough to justify doing a 3-PRP test, and only LL if that proves inconclusive. The repeated squaring of a Fermat test will only be slightly faster than LL, and 3-PRP Mersennes are probably more common than any saving we could produce. It would be nice to get some figures though. Chris Nash ------------------------------ From: Jean-Luc Cooke Date: Sun, 17 May 1998 19:31:37 -0400 Subject: Re: Mersenne: Use Fermat First? I've tried with Java and found it works for all the "small" Mersenne's. M(x), x <~ 2000. Not fast but seems right. TTYL JLC Will Edgington wrote: > > Chris Nash writes: > > I think there's a conjecture out there that if a Mersenne number of > prime exponent (and some other restriction) is pseudoprime to base > 3, then it is prime. Only a conjecture... > > This is related to - but not the same as - 'Conjecture B' of my > mersenne.html page. Peter-Lawrence Montgomery found a counter-example > for that, also in mersenne.html, most likely before I even thought of > the conjecture. > > Since that counter-example shows that factors of prime exponent > Mersenne numbers can be base-3 pseudo-primes, I suspect prime exponent > Mersennes can themselves be 3-PRP. As for how often, I don't have any > data, but writing a program to check using freeLIP would be > straight-forward and much faster than Java. > > Will > > http://www.garlic.com/~wedgingt/mersenne.html - -- Jean-Luc Cooke Carleton University, Electrical Engineering There is a 90% chance that right at this moment I am doing Calculus. God damn Newton, damn him to hell! - ---------------------------------------------------------------------- Hyperactivity Unbounded http://www.engsoc.carleton.ca/~jlcooke - ---------------------------------------------------------------------- AOL-IM ID: JLinOTTAWA http://register.netscape.oscar.aol.com - ---------------------------------------------------------------------- ICQ# 4474419 http://www.mirabilis.com - ---------------------------------------------------------------------- Email addresses jlcooke@ottawa.com jlcooke@magma.ca jlcooke@magmacom.com jlcooke@engsoc.carleton.ca jlcooke@chat.carleton.ca - ---------------------------------------------------------------------- ------------------------------ From: "Ernst W. Mayer" Date: Mon, 18 May 1998 14:36:32 -0400 Subject: Mersenne: Re: Optimising FFTs Nick Craig-Wood wriites: >It seems to me that you might be able to do an n/2 FFT of the non-zero bit >of the vector This can only be usefully exploited on the first pass. IMO, for any but the shortest FFT lengths, it's not worth it - you stand to gain much more from looking at memory layout, locality of data access and higher-radix implementations of the basic algorithm. >If you are not using the DWT (which I'm not because I'm working in an >integer field which doesn't support it) If you're doing Mersenne or Fermat-mod arithmetic in all integer, the DWT can be adapted to modular all-integer arithmetic. See Peter Montgomery's previous postings on this topic. Regards, Ernst ------------------------------ From: "Chuck W. " Date: Mon, 18 May 1998 11:53:31 -0700 (PDT) Subject: Re: Mersenne: Use Fermat First? I think the real question is, would the Fermat test ever say a number is "unlikely" to be prime when it is actually prime? The fermat test has to get us somewhere, hence we have to be able to eliminate exponents somehow. The real question is will the Fermat test give us any sort of consistent answers. If we were to test all prime exponents and the subset which was identified as "unlikely" to be prime, didn't contain any Mersenne primes, then we would have a mechanism for quick elimination. If however, there was even the smallest chance of an "unlikely to be prime" exponent, being prime, then I would say that it is not worth the time we would spend back tracking to double check ourselves. Having said all of that, can someone explain exactly how the Fermat test works? On Fri, 15 May 1998, Matt Daws wrote: > Dear all, > > I've kinda come to the same conclusion, but I think it should be a lot > faster than a LL test, as we (roughly) only have to do (p-1) squares, as > opposed to 2^(p-1) squares. Is there some reason why the Fermat test is > slow and/or would fail a lot of the time that we've missed?? > > Cheers for any info, > > Matt Daws > > ---------- > From: Jean-Luc Cooke > To: Mersenne Primes > Subject: Mersenne: Use Fermat First? > Date: 15 May 1998 20:39 > > Hey again, > > Sitting at home in the middle of a heat wave and I started think about > the LL test that I now finally understand. > > Q: Wouldn't it be faster to first test a number for pseudo primality > before going to the LL test? LL's take 50% more time than a Fermat test > and Fermat tests eliminate a BUNCH of candidates (i.e.. all but > Carmichel's). Just a 2-PRP test should do. After that's done go on to > the LL test. > > Or is this already being done before we get the numbers? It seems to me > that it isn't because it allot of work for the server to do. > > TTYL > > JLC > -- > Jean-Luc Cooke > Carleton University, Electrical Engineering > > There is a 90% chance that right at this moment I am doing Calculus. > God damn Newton, damn him to hell! > > ---------------------------------------------------------------------- > Hyperactivity Unbounded http://www.engsoc.carleton.ca/~jlcooke > ---------------------------------------------------------------------- > AOL-IM ID: JLinOTTAWA http://register.netscape.oscar.aol.com > ---------------------------------------------------------------------- > ICQ# 4474419 http://www.mirabilis.com > ---------------------------------------------------------------------- > Email addresses > jlcooke@ottawa.com > jlcooke@magma.ca jlcooke@magmacom.com > jlcooke@engsoc.carleton.ca jlcooke@chat.carleton.ca > ---------------------------------------------------------------------- > - - I tell you the truth, we speak of what we know, and we testify to what we have seen, but still you people do not accept our testimony. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ : WWW+PGP: http://www.silverlink.net/poke : : E-Mail: chuckw@silverlink.net : ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ : According to Section 227(b)((3)(B) of US Code Title 47 I am entitled : : to $500 per un-solicited commercial e-mail. : ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ------------------------------ From: Peter-Lawrence.Montgomery@cwi.nl Date: Mon, 18 May 1998 21:44:23 +0200 (MET DST) Subject: Re: Mersenne: Use Fermat First? Chuck W writes > I think the real question is, would the Fermat test ever say a number is > "unlikely" to be prime when it is actually prime? The fermat test has to > get us somewhere, hence we have to be able to eliminate exponents somehow. > The real question is will the Fermat test give us any sort of consistent > answers. The Fermat test will never reject true primes. If Mp = 2^p - 1 is prime, then a^(Mp + 1) == a^2 (mod Mp) for all bases a. One way to get consistent answers is to run it on bases a = 27 = 3^3 and a = 243 = 3^5. To test Mp, one Fermat tester repeatedly squares 27 until he gets 27^(Mp + 1). If his final result is 27^2 = 729, then Mp is a likely Mersenne prime, and an LL test can follow. Another tester computes 243^(Mp + 1). Even if both testers use the same program, they are putting different data through the FFT code. When done, the two testers report the bottom bits of [27^(Mp + 1)]^5 and [243^(Mp + 1)]^3 to a central site. The outcomes should agree, since both are computing (3^15)^(Mp + 1). [Actually one should use (Mp + 1)/2 rather than Mp as the exponent, checking whether the outcome is -27 or -243.] ------------------------------ From: Will Edgington Date: Mon, 18 May 1998 13:52:53 -0700 Subject: Mersenne: New mers format 'c:' line & beta mers release I've just updated my web pages, notably the data files. I wouldn't bother the entire list with this, since I've been doing it a few times a week lately, except that I've also added a new 'c:' line (and some other, smaller, improvements) to the mersfmt.txt file and to the beta ecm3.c program, which has new formulae for the starting ECM bound based on the table in the freeLIP documentation. The formulae use both the trial factoring extent and the prior "work" (see below or mersfmt.txt) and choose the larger starting bound. The beta mers release also now includes a new mers3p.c program to do 3-PRP tests of Mersenne numbers. I bothered with mers3p only because it took me less than an hour to modify my old program to work with rw.c's input(); I do not expect its speed to be anywhere near the speed of the LL test programs like mersenne1, if only because mers3p does not do any FFTs. I have started including the small (under 1200, so far, straight from the Cunningham Project data) composite exponent Mersenne numbers in the data and for P-1 factoring reservations. Note that the factors of a composite exponent Mersenne number, M(n) = 2^n - 1 where n is not prime, may be of the form k*n + 1 for odd k as well as for even k, so the trial factorers, including those of the mers package and of George Woltman's prime95, will miss about half of the factors. The ecm3 program of the mers package should not miss factors, nor should any of the other ECM programs nor any of the P-1 programs. The new 'c:' line records the number of digits of the composite cofactor when it was last checked by a pseudo-prime test like that of ecm3; e.g., the current data for M(601) is: M( 601 )C: 3607 M( 601 )C: 64863527 M( 601 )H: 1125904869540625 M( 601 )E: 17684960000 3000000 M( 601 )o: 12000000000 M( 601 )c: 170 The first two lines, with 'C:', are the known prime factors. The third, 'H:', line is the presumed extent of trial factoring; there may be gaps since my database update scripts still believe that all factors are from trial factoring except to try to record such gaps elsewhere. The fourth, 'E:', line is the ECM "work" and largest bound; the work is the sum of the bound over the curves performed, whether they find a factor or not. The fifth, 'o:', line is the P-1 stage one bound. The last and only new, 'c:', line indicates that the 170-digit cofactor was pseudo-prime tested and is known to be composite. If M(601) divided by the known factors were not 170 digits, then a new factor would have been discovered since the last pseudo-prime test and the cofactor might be prime and thus should be tested. One way to do this test is to run the new beta version of ecm3 with the new -p flag, which will do it if the digit count doesn't match or wasn't in the input. Will http://www.garlic.com/~wedgingt/mersenne.html http://www.garlic.com/~wedgingt/mersfmt.txt http://www.garlic.com/~wedgingt/beta.tgz ------------------------------ From: Jean-Luc Cooke Date: Mon, 18 May 1998 17:48:14 -0400 Subject: Re: Mersenne: Use Fermat First? Sure thing, The Fermat test looks like this: a^(N-1) mod N = 1 if N is prime. Or "newer" proofs have found that there are composite number that fail this test. These are called "Carmichel" numbers (spelling?). Not prime number fails these tests. At least I have NEVER heard or seen any. "a" can be any number really. But small primes are usually better. There are some theorems that say you can use combinations of fermat tests with various a's and be 100% confident of primality in certain ranges. Carmichel numbers are VERY rare. The only Carmichel number under 100,000 are: 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, and 75361. That's 16/100,000 = 0.016% and there are only 2,163 under 25,000,000. Mersenne numbers as I have need shown a while ago are Fermat test prime with a=2 (or more commonly written 2-PRP) for M(n) = 2^n - 1, if n is prime. So there it is. Now the question I started asking was that if we used a Fermat test BEFORE the LL test to look for primes. 2-PRP is unless because the primeNet server only sends us prime exponents (see above why this is no good). So I just fell back on the question that we should try 3,5,7, or what ever x-PRP to test. Chris Nash and I are waiting for the PrimeNet/GIMPS programmers/mathy-people to step in and clear this up. Is the Fermat test a good pre tester seeing how no prime will me "missed" and there is a VERY-VERY-VERY rare change that we'd pick up a composite (37 +/- 5 Mersenne's under a goggle, how many are Carmichels?) and on that very odd chance that a Mersenne number passes the Fermat test, there is no composite number known to fail both Fermat and LL. Well that's as I see it. If I've made any errors, I apologize my mis-quoting someone or a source. TTYL JLC refs: http://www.utm.edu/research/primes/prove2.html#prp Chirs Nash And all you other people! Chuck W. wrote: > > I think the real question is, would the Fermat test ever say a number is > "unlikely" to be prime when it is actually prime? The fermat test has to > get us somewhere, hence we have to be able to eliminate exponents somehow. > The real question is will the Fermat test give us any sort of consistent > answers. > > If we were to test all prime exponents and the subset which was identified > as "unlikely" to be prime, didn't contain any Mersenne primes, then we > would have a mechanism for quick elimination. If however, there was even > the smallest chance of an "unlikely to be prime" exponent, being prime, > then I would say that it is not worth the time we would spend back > tracking to double check ourselves. > > Having said all of that, can someone explain exactly how the Fermat test > works? > > On Fri, 15 May 1998, Matt Daws wrote: > > > Dear all, > > > > I've kinda come to the same conclusion, but I think it should be a lot > > faster than a LL test, as we (roughly) only have to do (p-1) squares, as > > opposed to 2^(p-1) squares. Is there some reason why the Fermat test is > > slow and/or would fail a lot of the time that we've missed?? > > > > Cheers for any info, > > > > Matt Daws > > > > ---------- > > From: Jean-Luc Cooke > > To: Mersenne Primes > > Subject: Mersenne: Use Fermat First? > > Date: 15 May 1998 20:39 > > > > Hey again, > > > > Sitting at home in the middle of a heat wave and I started think about > > the LL test that I now finally understand. > > > > Q: Wouldn't it be faster to first test a number for pseudo primality > > before going to the LL test? LL's take 50% more time than a Fermat test > > and Fermat tests eliminate a BUNCH of candidates (i.e.. all but > > Carmichel's). Just a 2-PRP test should do. After that's done go on to > > the LL test. > > > > Or is this already being done before we get the numbers? It seems to me > > that it isn't because it allot of work for the server to do. > > > > TTYL > > > > JLC > > -- > > Jean-Luc Cooke > > Carleton University, Electrical Engineering > > > > There is a 90% chance that right at this moment I am doing Calculus. > > God damn Newton, damn him to hell! > > > > ---------------------------------------------------------------------- > > Hyperactivity Unbounded http://www.engsoc.carleton.ca/~jlcooke > > ---------------------------------------------------------------------- > > AOL-IM ID: JLinOTTAWA http://register.netscape.oscar.aol.com > > ---------------------------------------------------------------------- > > ICQ# 4474419 http://www.mirabilis.com > > ---------------------------------------------------------------------- > > Email addresses > > jlcooke@ottawa.com > > jlcooke@magma.ca jlcooke@magmacom.com > > jlcooke@engsoc.carleton.ca jlcooke@chat.carleton.ca > > ---------------------------------------------------------------------- > > > > - > I tell you the truth, we speak of what we know, and we testify to what > we have seen, but still you people do not accept our testimony. > > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > : WWW+PGP: http://www.silverlink.net/poke : > : E-Mail: chuckw@silverlink.net : > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > : According to Section 227(b)((3)(B) of US Code Title 47 I am entitled : > : to $500 per un-solicited commercial e-mail. : > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -- Jean-Luc Cooke Carleton University, Electrical Engineering There is a 90% chance that right at this moment I am doing Calculus. God damn Newton, damn him to hell! - ---------------------------------------------------------------------- Hyperactivity Unbounded http://www.engsoc.carleton.ca/~jlcooke - ---------------------------------------------------------------------- AOL-IM ID: JLinOTTAWA http://register.netscape.oscar.aol.com - ---------------------------------------------------------------------- ICQ# 4474419 http://www.mirabilis.com - ---------------------------------------------------------------------- Email addresses jlcooke@ottawa.com jlcooke@magma.ca jlcooke@magmacom.com jlcooke@engsoc.carleton.ca jlcooke@chat.carleton.ca - ---------------------------------------------------------------------- ------------------------------ From: Jean-Luc Cooke Date: Mon, 18 May 1998 18:23:17 -0400 Subject: Re: Mersenne: Use Fermat First? Curious, I'm not clear why this is the case? So you're saying that (1) a^( [Mp + 1]/2 ) mod Mp = -a ?? Thank you. If this is faster than (2) a^(Mp+1) mod Mp = 1 Than we should DEFINATLY use the Fermat test before the LL!!! I'm using (2) and getting 33% fast CPU time. If I use (1) I'll get dramaticly faster times still!!! TTYL JLC Peter-Lawrence.Montgomery@cwi.nl wrote: > [Actually one should use (Mp + 1)/2 rather than Mp as the exponent, > checking whether the outcome is -27 or -243.] - -- Jean-Luc Cooke Carleton University, Electrical Engineering There is a 90% chance that right at this moment I am doing Calculus. God damn Newton, damn him to hell! - ---------------------------------------------------------------------- Hyperactivity Unbounded http://www.engsoc.carleton.ca/~jlcooke - ---------------------------------------------------------------------- AOL-IM ID: JLinOTTAWA http://register.netscape.oscar.aol.com - ---------------------------------------------------------------------- ICQ# 4474419 http://www.mirabilis.com - ---------------------------------------------------------------------- Email addresses jlcooke@ottawa.com jlcooke@magma.ca jlcooke@magmacom.com jlcooke@engsoc.carleton.ca jlcooke@chat.carleton.ca - ---------------------------------------------------------------------- ------------------------------ End of Mersenne Digest V1 #363 ******************************