Mersenne Digest Friday, 15 May 1998 Volume 01 : Number 361 ---------------------------------------------------------------------- From: Nick Craig-Wood Date: Thu, 14 May 1998 20:41:27 +0100 (BST) Subject: Mersenne: Optimising FFTs A question for all you FFT experts out there! If you are not using the DWT (which I'm not because I'm working in an integer field which doesn't support it) you start off doing an FFT of a vector length n which is half 0s. It seems to me that you might be able to do an n/2 FFT of the non-zero bit of the vector then combine this with another FFT-like pass to get the full length n vector, saving yourself about 1/2 the time. Eg :- Original :- ( a b c d 0 0 0 0 ) FFT ( A B C D E F G H ) Proposed :- ( a b c d ) FFT ( X Y Z T ) An extra (magic!) pass ( A B C D E F G H ) It is easy to save 1/2 a round at the start (or even the whole first round) but more seems difficult. - -- Nick Craig-Wood ncw1@axis.demon.co.uk http://www.axis.demon.co.uk/ ------------------------------ From: "John R Pierce" Date: Thu, 14 May 1998 14:00:02 -0700 Subject: Re: Mersenne: Optimising FFTs But a 256k element FFT takes something like 18 passes. Saving half of one pass isnt much (assuming even that much can be done). - -----Original Message----- From: Nick Craig-Wood To: Mersenne List Date: Thursday, May 14, 1998 1:30 PM Subject: Mersenne: Optimising FFTs >A question for all you FFT experts out there! > >If you are not using the DWT (which I'm not because I'm working in an >integer field which doesn't support it) you start off doing an FFT of a >vector length n which is half 0s. > >It seems to me that you might be able to do an n/2 FFT of the non-zero bit >of the vector then combine this with another FFT-like pass to get the full >length n vector, saving yourself about 1/2 the time. Eg :- > >Original :- > ( a b c d 0 0 0 0 ) >FFT > ( A B C D E F G H ) > >Proposed :- > ( a b c d ) >FFT > ( X Y Z T ) >An extra (magic!) pass > ( A B C D E F G H ) > >It is easy to save 1/2 a round at the start (or even the whole first round) >but more seems difficult. > >-- >Nick Craig-Wood >ncw1@axis.demon.co.uk >http://www.axis.demon.co.uk/ > > ------------------------------ From: Jean-Luc Cooke Date: Thu, 14 May 1998 23:53:37 -0400 Subject: Mersenne: LL Tests... This is a multi-part message in MIME format. - --------------F2A381217B57152CD42A32A3 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Hi again, At http://www.utm.edu/research/primes/mersenne.shtml I saw a quick explaination of the LL test. I'm new to N.T. and I'd just like a clearer explanation. For p odd, the Mersenne number 2p-1 is prime if and only if 2p-1 divides S(p-1) where S(n+1) = S(n)^2 - 2, and S(1) = 4. So for example: p S(p-1) 3 4^2-2 = 14 5 14^2-2 = 194 . . . . . . I this how it's saposed to work? I'm a bit perplexed about how to implement the algorithm. 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 - ---------------------------------------------------------------------- - --------------F2A381217B57152CD42A32A3 Content-Type: text/x-vcard; charset=us-ascii; name="vcard.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Jean-Luc Cooke Content-Disposition: attachment; filename="vcard.vcf" 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 - --------------F2A381217B57152CD42A32A3-- ------------------------------ From: Simon Burge Date: Fri, 15 May 1998 15:53:57 +1000 Subject: Re: Mersenne: LL Tests... On Thu, 14 May 1998 23:53:37 -0400 Jean-Luc Cooke wrote: > At http://www.utm.edu/research/primes/mersenne.shtml I saw a quick > explaination of the LL test. I'm new to N.T. and I'd just like a > clearer explanation. > > For p odd, the Mersenne number 2p-1 is prime if and only if 2p-1 divides > S(p-1) where S(n+1) = S(n)^2 - 2, and S(1) = 4. > > So for example: > > p S(p-1) > > 3 4^2-2 = 14 > 5 14^2-2 = 194 > . . > . . > . . > > I this how it's saposed to work? I'm a bit perplexed about how to > implement the algorithm. It's an iterative algorithm. For example, to check M5 (p = 5), start with s = 4 (as always), and loop from 3 to p: s = 4 iter #3: s = 4^2-2 mod (2^5-1) = 16-2 mod 31 = 14 mod 31 = 14 iter #4: s = 14^2-2 mod (2^5-1) = 196-2 mod 31 = 194 mod 31 = 8 iter #5: s = 8^2-2 mod (2^5-1) = 64-2 mod 31 = 62 mod 31 = 0 At the end s == 0, so M5 is prime. If s were any other value, the exponent would not have been prime. Simon. ------------------------------ From: Nick Craig-Wood Date: Fri, 15 May 1998 08:46:19 +0100 (BST) Subject: Re: Mersenne: Optimising FFTs In on Thu 14 May, John R Pierce wrote: > But a 256k element FFT takes something like 18 passes. Saving half of one > pass isnt much (assuming even that much can be done). That is all I've worked out how to do. Hence my posting - is it possible to do a 128k FFT on the non-zero data and then do one extra pass to bring it up to 256k. This would bring a considerable saving (in the non-DWT case). - -- Nick Craig-Wood ncw1@axis.demon.co.uk http://www.axis.demon.co.uk/ ------------------------------ From: "Chuck W. " Date: Fri, 15 May 1998 11:00:35 -0700 (PDT) Subject: Re: Mersenne: LL Tests... Why do we loop from 3 to p??? Why not from 0 to p or 1 to p??? On Fri, 15 May 1998, Simon Burge wrote: > On Thu, 14 May 1998 23:53:37 -0400 Jean-Luc Cooke wrote: > > > At http://www.utm.edu/research/primes/mersenne.shtml I saw a quick > > explaination of the LL test. I'm new to N.T. and I'd just like a > > clearer explanation. > > > > For p odd, the Mersenne number 2p-1 is prime if and only if 2p-1 divides > > S(p-1) where S(n+1) = S(n)^2 - 2, and S(1) = 4. > > > > So for example: > > > > p S(p-1) > > > > 3 4^2-2 = 14 > > 5 14^2-2 = 194 > > . . > > . . > > . . > > > > I this how it's saposed to work? I'm a bit perplexed about how to > > implement the algorithm. > > It's an iterative algorithm. For example, to check M5 (p = 5), start > with s = 4 (as always), and loop from 3 to p: > > s = 4 > > iter #3: s = 4^2-2 mod (2^5-1) = 16-2 mod 31 = 14 mod 31 = 14 > iter #4: s = 14^2-2 mod (2^5-1) = 196-2 mod 31 = 194 mod 31 = 8 > iter #5: s = 8^2-2 mod (2^5-1) = 64-2 mod 31 = 62 mod 31 = 0 > > At the end s == 0, so M5 is prime. If s were any other value, the > exponent would not have been prime. > > Simon. > - - 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: Jean-Luc Cooke Date: Fri, 15 May 1998 15:39:24 -0400 Subject: Mersenne: Use Fermat First? This is a multi-part message in MIME format. - --------------4B767E215623B6DE3CA32535 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit 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 - ---------------------------------------------------------------------- - --------------4B767E215623B6DE3CA32535 Content-Type: text/x-vcard; charset=us-ascii; name="vcard.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Jean-Luc Cooke Content-Disposition: attachment; filename="vcard.vcf" 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 - --------------4B767E215623B6DE3CA32535-- ------------------------------ From: "Matt Daws" Date: Fri, 15 May 1998 23:18:43 +0100 Subject: Re: Mersenne: Use Fermat First? 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 - ---------------------------------------------------------------------- ------------------------------ From: Jean-Luc Cooke Date: Fri, 15 May 1998 18:34:17 -0400 Subject: Re: Mersenne: Use Fermat First? This is a multi-part message in MIME format. - --------------7C452ED9A683173F0F249E3C Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Humm... Nope. I just tried it with the first 1000, not all (any?) "2^p - 1" are pseudo (Fermat) primes. Though it would have explained things if it had!!! TTYL JLC two inch deep pile carpet wrote: > > Jean-Luc Cooke wrote: > > > > 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 > > I don't understand ANY of the math involved, but I recall (tell me > if I haven't got this rememberized correctly) that mersenne numbers > categorically pass the Fermat test. > > ______________________________________________________________________ > David Nicol 816.235.1187 UMKC Network Operations david@news.umkc.edu > "Irving was twirling his gun around..." - -- 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 - ---------------------------------------------------------------------- - --------------7C452ED9A683173F0F249E3C Content-Type: text/x-vcard; charset=us-ascii; name="vcard.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Jean-Luc Cooke Content-Disposition: attachment; filename="vcard.vcf" 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 - --------------7C452ED9A683173F0F249E3C-- ------------------------------ From: Chris Nash Date: Fri, 15 May 1998 19:04:46 -0400 Subject: Re: Mersenne: Use Fermat First? Jean-Luc >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. This was a theory I believed for a while as well. However it doesn't hold up. Firstly, the LL test takes 'practically' the same amount of time as a Fermat test. The implementation just involves repeating squarings - I guess your 50% comes from considering a^((n-1)/2), so to evaluate this wouldn't require 50% of the time, but would just miss off the last iteration. (The only difference in the LL test is that each squaring is followed by a subtraction, which apparently is inexpensive). So, *at the moment*, a Fermat test isn't a great filter. It takes just as long and is likely to give an inconclusive result. Unless there's a much better algorithm for Fermat than the one that's comparable to LL, you may as well just use LL and get a definite result. By the way, all Mersenne numbers with prime exponent are pseudoprime to base 2. Not very difficult to prove, apparently... if m=2^p-1, then m=1 modulo p. So 2^(m-1)=(2^kp) for some k. And so 2^(m-1) = (2^p)^k = 1^k =1 modulo m. Chris Nash ------------------------------ From: Simon Burge Date: Sat, 16 May 1998 09:26:18 +1000 Subject: Re: Mersenne: LL Tests... On Fri, 15 May 1998 11:00:35 -0700 (PDT) "Chuck W. " wrote: > Why do we loop from 3 to p??? Why not from 0 to p or 1 to p??? Um, because that's what the algorith says to do? Honestly, I do not know the reason, but it works... Simon. ------------------------------ From: Jean-Luc Cooke Date: Fri, 15 May 1998 20:47:15 -0400 Subject: Re: Mersenne: Use Fermat First? This is a multi-part message in MIME format. - --------------DE734DCAF2A6ACC51217BC86 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Chris Nash wrote: > > Jean-Luc > > >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. > > This was a theory I believed for a while as well. However it doesn't hold > up. > > Firstly, the LL test takes 'practically' the same amount of time as a Fermat > test. The implementation just involves repeating squarings - I guess your > 50% comes from considering a^((n-1)/2), so to evaluate this wouldn't require > 50% of the time, but would just miss off the last iteration. (The only > difference in the LL test is that each squaring is followed by a > subtraction, which apparently is inexpensive). Humm, I'm using 2^(N-1) mod N = 1 for primality. It returns a true/false in, say, 6 seconds. The LL test returns an answer in, say, 12 seconds. It will not CONFIRM primality, but it will eliminate composites. I'm afraid I don't see how this takes the same amount of time as LL? If a number passes 2^(N-1) mod N = 1, then we can test by LL to be 100% sure instread of 99.9995% sure. I think a savings of 33% is significant enough. Perhaps Fermat/LL ratio depreciates with size, but I havn't seen that trend on my 'puter. - -- 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 - ---------------------------------------------------------------------- - --------------DE734DCAF2A6ACC51217BC86 Content-Type: text/x-vcard; charset=us-ascii; name="vcard.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Jean-Luc Cooke Content-Disposition: attachment; filename="vcard.vcf" 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 - --------------DE734DCAF2A6ACC51217BC86-- ------------------------------ From: Chris Nash Date: Fri, 15 May 1998 21:23:38 -0400 Subject: Re: Mersenne: Use Fermat First? Matt >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?? *scratch*. An LL test only does (p-1) (or thereabouts) squarings as well. Each iteration of the LL test is a squaring and a subtraction, and it takes (p-1) (or thereabouts) iterations to primality test a Mersenne prime candidate. Chris Nash ------------------------------ From: Chris Nash Date: Fri, 15 May 1998 21:49:51 -0400 Subject: Re: Mersenne: Use Fermat First? >Humm, I'm using 2^(N-1) mod N = 1 for primality. It returns a >true/false in, say, 6 seconds. The LL test returns an answer in, say, >12 seconds. It will not CONFIRM primality, but it will eliminate >composites. I'm afraid I don't see how this takes the same amount of >time as LL? *smiles* Neither could I at first. These sound like reasonably small N. But consider what happens when N is a large Mersenne number with prime exponent p, N=2^p-1. You might as well be calculating 2^(2^p) and testing for equality to 4 modulo N. So that takes p squarings, at each stage you're working with a p-bit number. For LL, you need p-1 iterations (I think, it may be p). Each is a squaring, followed by a subtract 2. Even with a comventional algorithm, the subtract 2 is dwarved by the effort required for the squaring and the reduction modulo N, because p is fairly large. > I think a savings of 33% is significant Possibly. Provided you could choose a base for pseudo-primality testing that would confirm a candidate composite more often than not - and the saving of Fermat was significant enough. I doubt Fermat saves much, and certainly not enough to justify risking needing to double the test time by having to follow with LL. Chris Nash ------------------------------ From: Jean-Luc Cooke Date: Fri, 15 May 1998 21:57:45 -0400 Subject: Re: Mersenne: Use Fermat First? This is a multi-part message in MIME format. - --------------C71A66875E1B24D09A6AC9F5 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit OKAY, http://www.engsoc.carleton.ca/~jlcooke/Numbers/GIMPS/ll.html I'm not discrediting what everyone's doing here. But I've hacked up an ugly applet that tests using LL and an Eratosthenes/Fermat combonation. If you count in your head you can see that it's a bit faster that LL. Like I said before Ferm:LL = 2:3. Try higher numbers if you'd like. But remember.. it's Java and not NEARLY as CPU thrashing as Prime95. Kinda cool really, I turn my speekers up and listen to Java go at it. It's dull buzzing. When it switches back to Prime95 it's like a crikket chirping. Beautiful! TTYL JLC Chris Nash wrote: > > >Humm, I'm using 2^(N-1) mod N = 1 for primality. It returns a > >true/false in, say, 6 seconds. The LL test returns an answer in, say, > >12 seconds. It will not CONFIRM primality, but it will eliminate > >composites. I'm afraid I don't see how this takes the same amount of > >time as LL? > > *smiles* Neither could I at first. These sound like reasonably small N. But > consider what happens when N is a large Mersenne number with prime exponent > p, N=2^p-1. You might as well be calculating 2^(2^p) and testing for > equality to 4 modulo N. So that takes p squarings, at each stage you're > working with a p-bit number. > > For LL, you need p-1 iterations (I think, it may be p). Each is a squaring, > followed by a subtract 2. Even with a comventional algorithm, the subtract 2 > is dwarved by the effort required for the squaring and the reduction modulo > N, because p is fairly large. > > > I think a savings of 33% is significant > > Possibly. Provided you could choose a base for pseudo-primality testing that > would confirm a candidate composite more often than not - and the saving of > Fermat was significant enough. I doubt Fermat saves much, and certainly not > enough to justify risking needing to double the test time by having to > follow with LL. > > Chris Nash - -- 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 - ---------------------------------------------------------------------- - --------------C71A66875E1B24D09A6AC9F5 Content-Type: text/x-vcard; charset=us-ascii; name="vcard.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Jean-Luc Cooke Content-Disposition: attachment; filename="vcard.vcf" 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 - --------------C71A66875E1B24D09A6AC9F5-- ------------------------------ End of Mersenne Digest V1 #361 ******************************