Re: Mersenne: pi
On Fri, 11 Feb 2000 [EMAIL PROTECTED] wrote: > It is my understanding that this list is for the promotion of the search > for mersenne primes. I may be a laymen when it comes to mathematics, but to > my knowlege, this is not a requirement of the list. ... > I would hope that your opinion is in the minority. It smacks of elitieism. Ouch. Sorry, there. Perhaps I misspoke; I was certainly misunderstood. I applaud anyone who is interested in math, and this mailing list is certainly full of such people. We often have good conversations here about all sorts of topics related to math, pi, infinities, whatever; last I recall we have Math PHD's and, well, cabinetmakers on the list and anyone in between. All welcome, and while it's not my place to do so, I encourage people to post; the archives are rich with posts on every topic and at every level of math. My apologies, anyway, if I sounded off the mark. I meant no ill will, and I believe I mentioned how enthusiastic I was about the thread. I just thought there may be better resources than here for certain discussions. Not so much that they don't belong here than that your questions may well be better answered somewhere else. I was, unbelievably, trying to be helpful. Anyway, sorry again. It shan't happen again. ---Chip \\ ^ // (o o) ---oOO--(_)--OOo | Chip Lynch| Computer Guru| | [EMAIL PROTECTED] || | (703) 465-4176 (w) | (202) 362-7978 (h) | _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
> Circumference/diameter is a ratio. The decimal value 3.1415-> ... > Seems to me a ratio is needed for pi. you already gave it, the 'ratio of pi' is circumference / diameter There are NO two integers which can divide to create PI. Thats why PI is considered a 'irrational' number. It has no simple ratio. _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Mersenne Digest Archives
Is there an accessible archive of Mersenne Digests written in the past year+. I may be using an old web address. The latest archived issue that I can find is dated August 1998 (number 408). Irv Rosenfeld _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: pi
Disclaimer: I am not a mathematician. It is a hobby. I am a cabinetmaker. At 03:19 PM 2/11/00 -0600, Jeremy Blosser wrote: >I think the mistake you are making is that the *precision* of PI is infinite >(never ending), but PI itself is not "infinity". > >3.14159.->ininite number of numbers > >Since it is 3.something we know it is > 3 and < 4. > >Take 1/3 for example. > >Its decimal value us 0.3-> infinte # of 3's > >But it is < 3 and > 0. Just because the number of 3's in it is infinite does >not mean it is "infinity". > 1/3 is a ratio. The decimal value .333..-> Circumference/diameter is a ratio. The decimal value 3.1415-> !/3 of a 12 inch rope is 4 inch d/c of a 12 inch rope is ? (3.1415..->something * 12) Seems to me a ratio is needed for pi. Dan- _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
At 04:51 PM 2/11/00 -0500, Chip Lynch wrote: >Language, NOT Mathematics, is (precisely) why these discussions are >problematic. If you've ever read original works by Archimedes, Euclid, >and others who try to define mathematics with a common language, you >understand the frustration. > >While I think the topic is stimulating and important, the Mersenne list >probably isn't the best medium for it, unfortunately. Anyone recommend a >few good links on the subject? (Pi, The Language of Mathematics, any of >that) > >---Chip > Disclaimer: I am not a mathematician. It is a hobby. I am a cabinetmaker. It is my understanding that this list is for the promotion of the search for mersenne primes. I may be a laymen when it comes to mathematics, but to my knowlege, this is not a requirement of the list. I believe that pi and mersenne prime numbers may have a relationship. I could be and might be wrong. I have tried three times now to put forth my observations on the mersenne list, but for some reason, my pi-4 posts are not being posted. All my other posts are posted on the list. The argument I propose, in this illusive post, is partly observation, facts, and a Biblical metaphysical outlook of the world. I have contributed to the GIMPS project and I am interested in furthering the search. I have and continue to learn much from the many posts. I enjoy logging on and reading them. I must admit that I have been bored by some of the long lists of prime95 primesearch programming and implementation posts about different computers etc. But I realise these play their part too. My opinion is that just because we have prime95 doesn't mean that a better way will not be found at some point. I may not be able to speak in "state of the art mathematical terms" but as I become more educated by participating in these discussions, I may be able to contribute something substantial. Perhaps not. These posts increase and hold my enthusiasm for math and mersennes. I hope this enthusiasm spreads. I would hope that your opinion is in the minority. It smacks of elitieism. I have admired the understanding people have shown me in replies to mine and other posts. I bear no grudge. I am just trying to stand on the shoulders of Giants. Dan- _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
> >> Infinite to me means never ending. A precisely defined value to me is a > >> finite value. > > > > Your definition of infinite is not correct. > > Just glanced at my Websters Dictionary. infinite: 1. lacking limits; endless. > Endless and never ending seem synonymous to me. > What dictionary are you using? > > >> A never ending value is not finite. > > > > Pi is greater than 3 and less than 4, therefore it is finite. I don't mind people using their own definitions of words; it's a shortcoming of mathematics that fundamentally needs improvement. However, once definitions are agreed upon, they must be at least internally consistent. If you take finite and infinite to be exact opposites, then we can't discuss the magnitude of pi (the fact that it is greater than three and less than four) to mean its "finite"ness, and it's length in decimal digits to mean its "infinite"ness. If we did, it could be both finite and infinite at the same time which is apocryful if those are opposites. Language, NOT Mathematics, is (precisely) why these discussions are problematic. If you've ever read original works by Archimedes, Euclid, and others who try to define mathematics with a common language, you understand the frustration. While I think the topic is stimulating and important, the Mersenne list probably isn't the best medium for it, unfortunately. Anyone recommend a few good links on the subject? (Pi, The Language of Mathematics, any of that) ---Chip \\ ^ // (o o) ---oOO--(_)--OOo | Chip Lynch| Computer Guru| | [EMAIL PROTECTED] || | (703) 465-4176 (w) | (202) 362-7978 (h) | _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Fwd: Re: Mersenne: Re: Base e arithmetic
>This led to a discussion as to whether or not it is possible to have a number >system based on a non-integer base. Maybe the great minds of GIMPS >can contribute to this. Base phi is easy to compute in. The similar Fibonacci representation counts the integers as follows: 0,1,10,100,101,1000,1001,1010,1,10001,10010,10100,10101... No number has two 1-bits adjacent. Complex bases can also be used. i-1 is usable as a base, but i+1 is not, because 2 cannot be finitely represented. Gosper's representation uses seven digits, the sixth roots of 1 and 0, in base 2.5+sqrt(-3/4). phma _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: pi
I think the mistake you are making is that the *precision* of PI is infinite (never ending), but PI itself is not "infinity". 3.14159.->ininite number of numbers Since it is 3.something we know it is > 3 and < 4. Take 1/3 for example. Its decimal value us 0.3-> infinte # of 3's But it is < 3 and > 0. Just because the number of 3's in it is infinite does not mean it is "infinity". _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Re: irrational bases
George Sassoon ([EMAIL PROTECTED]) wrote (regarding base-e arithmetic): >Maybe pi could be expressed exactly in such a >system. After all, e^(pi*i) = -1. Indeed, it follows that pi=-i*log(-1). But now one has the problem that one has defined one transcendental number in terms of another, and is thus no closer to writing pi in closed form in terms of whole numbers. I prefer to use base i, i.e. to no longer restrict oneself to the reals. In the complex (x,y) plane (i.e. every complex number x + i*y is viewed as the point (x,y) in a Cartesian-style coordinate system), one has that i = (0,1), which involves just integers. One also has the nifty identity i^i = -pi/2 i.e. the imaginary number i, raised to itself, gives - voila!- a real result involving pi in a simple way. (Proof: log(i^i) = i*log(i). But in complex polar form, i = e^(i*pi/2), so log(i) = i*pi/2, from which the result quickly follows.) Thus, pi=-2*i^i. >This led to a discussion as to whether or not it is possible to have a number >system based on a non-integer base. Your example with base e shows that indeed, one can. On the other hand, i (as in my example) is not an eligible base for a number system, since it has complex modulus one (i.e. one can raise i to any power one likes but never get off the unit circle.) But one can simply use, say, b = 2*i as the base, and then one has the identity pi = -2*(b/2)^(b/2). And now, at this point of our seemingly off-topic foray into the wonders of pi, e and the complex numbers, we suddenly are back on-topic. Here's why: Richard Crandall, who devised DWT algorithm at the heart of the Mersenne testing codes all of use in one form or another, refers to it, in words, as the "irrational-base discrete weighted transform" (IBDWT). I've quibbled with him about his choice of wording here, since it seems to cause much confusion among folks who have not actually implemented such an algorithm computationally. The source of the confusion is that, even though representing a prime-exponent Mersenne number 2^p-1 using N binary words is in some sense mathematically equivalent to using an irrational-base number system, in practice one uses a mixture of base 2^s and 2*2^s, where s = floor(p/N), i.e. is an integer. Thus, one doesn't actually use irrational bases anywhere in the actual algorithm. -Ernst _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: pi
At 01:14 PM 2/11/00 -0600, Kyle Evans wrote: > >> I have a circle with a area of 5 square inches drawn on my pad. 5 inches >is >> precise. > >You do?? How did you do that? Did you set your compass so that the points >were exactly sqrt (5/pi) inches apart? > >What you probably have really is a circle that approximates 5 sq inches >enough to suit YOU. > >Kyle Evans. > I just drew a circle. Then wrote inside the circle: "The area is precisely 5 square inches". Dan _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: NTPrime and Prime9x simultaneously ?
On Fri, 11 Feb 2000, Alex Phillips wrote: >Dear All, > Can anyone tell me if it is possible to have both the NT Prime service >(under NT) and Prime95 (Under Win98), both installed in the same directory, >and sharing files on a dual boot machine (NT for work and 98 for games). My guess is yes, as long as the executables have different names. I have had mprime and Prime95 running in the same directory, one picking up where the other left off. > Because I can't seem to get the NT setup program to report on it's status. >So I don't know if it's working ? Look for a file with a name beginning with a p and ending in a bunch of numbers. It should be the last file when sorted by mtime. If the program is running, it will write to that file at intervals. phma _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Pi and Greek
>Date: Fri, 11 Feb 2000 14:32:48 -0600 >To: "Vincent J. Mooney Jr." <[EMAIL PROTECTED]> >From: [EMAIL PROTECTED] >Subject: Re: Mersenne: Pi and Greek >In-Reply-To: <[EMAIL PROTECTED]> > >Disclaimer: I am not a mathematician. It is a hobby. I am a cabinetmaker. > >I was not necessarily speaking in a factual, but a Biblical, metaphysical manner. >Dan- > >At 10:56 PM 2/10/00 -0500, you wrote: >>Not so. Read up on why pi was selected, by whom and when. >> >>History of Pi by Peter Beckmann is one source. >> >>At 02:58 PM 2/10/00 -0600, Dan wrote: >>>I just found out that pi is the 16th letter of the Greek Alphabet. So this >>>is more evidence for the theory that the numbers 4 and 16 are important in >>>regards to pi and also to mersenne primes. >>>Dan >> > _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: NTPrime and Prime9x simultaneously ?
From: Alex Phillips > Can anyone tell me if it is possible to have both > the NT Prime service (under NT) and Prime95 (Under > Win98), both installed in the same directory, and > sharing files on a dual boot machine (NT for work > and 98 for games). That's a definite 'yes' Alex. Just make sure that you use the same versions of both programs and upgrade them at the same time when the time comes. I did that for a long time and both programs co-exist in the same directory and use the same data and ini files with no trouble at all. Rick. -+--- [EMAIL PROTECTED] http://www.alienshore.com/ _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
At 11:39 AM 2/11/00 -0600, you wrote: > > >On Fri, 11 Feb 2000 [EMAIL PROTECTED] wrote: > >> Infinite to me means never ending. A precisely defined value to me is a >> finite value. > > Your definition of infinite is not correct. Just glanced at my Websters Dictionary. infinite: 1. lacking limits; endless. Endless and never ending seem synonymous to me. What dictionary are you using? >> A never ending value is not finite. > > Pi is greater than 3 and less than 4, therefore it is finite. > > Question to you. How much greater is pi to 3 or less than 4. If you cannot give me a answer then I will help you. Subtract 3 from your "finite" value of pi, or subtract your "finite" value of pi from 4. No approximations. Please give the answer in finite terms. Disclaimer: All email I have sent to the [EMAIL PROTECTED] list since 2-9-00 assumed that all on the list had read my post "Re: more info on pi" in which I explained that "I am not a mathematician." . However: that post seems to have disapeared. Then I changed the title to "Re: pi and 4" and re-sent again at 9:38 A.m. Central time 2-11-00 (this morning). cc'ing to [EMAIL PROTECTED] as per his offer to help in tracking it. This email will give all a very good understanding of my observations if it should ever be delivered. I am contemplating re-sending it again. I am not sure why? Remember; I am not a mathematician. It is a hobby. I am a cabinetmaker. And also, you know what happens when you assume? Well, perhaps, I have made one out of you and me. :) smile Good natured.. thats me. Really. Thanks for bearing up under my verse. Dan- _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Where's the flaw in my thinking?
>If we left shift this by the position of the 1, for each 1 in the binary >representation, and add them together, we should get the square... So to >square 14, we do this: >1110 << 3 == 111 + >1110 << 2 == 0111000 + >1110 << 1 == 0011100 + > == 11000100 which is 196 > >So for each squaring, we have x left shifts and adds, where x is no larger >that p. > >In any case... is this just me being dumb and missing that this is just a >stupid way of squaring a number? It is not stupid, it is in fact the normal way of squaring a small number. The reason humongous numbers are squared with FFT is that shift-and-add takes O(n^2) bit operations, where n is the number of bits in the number being squared, whereas FFT takes O(n*ln(n)*ln(ln(n))), which is a lot faster. The FFT of n floats takes log_2(n) stages, each of which runs through the array once which takes n operations; the ln(ln(n)) term (please correct me if this term is wrong) is because the more bits there are in the number you are squaring, the more precisely you need to compute the floats in the FFT. phma _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: NTPrime and Prime9x simultaneously ?
Dear All, Can anyone tell me if it is possible to have both the NT Prime service (under NT) and Prime95 (Under Win98), both installed in the same directory, and sharing files on a dual boot machine (NT for work and 98 for games). Because I can't seem to get the NT setup program to report on it's status. So I don't know if it's working ? Any Ideas ? Kind Regards, Alex Phillips. _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Where's the flaw in my thinking?
At 12:23 PM 2/11/00 -0600, you wrote: >Okay, I was sitting there the other day thinking about a non-FFT squaring >algorithm... > >Say we have 14, which in binary is 1110... > >If we left shift this by the position of the 1, for each 1 in the binary >representation, and add them together, we should get the square... So to >square 14, we do this: >1110 << 3 == 111 + >1110 << 2 == 0111000 + >1110 << 1 == 0011100 + > == 11000100 which is 196 > In other words you are saying 14*14 = 14*(8 + 4 + 2) = 14*8 + 14*4 + 14*2 or in base 10 14*14 = 14*(10 + 4) = 14*10 + 14*4 Your logic appears correct, but it appears to be a computer base 2 implementation of what we do longhand in base 10. You are just taking advantage of the fact that multiplying by 2 is base 2 is a position shift, just like it is real easy in base 10 for us to take 1234 * 100 = 123400 _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: pi
> I have a circle with a area of 5 square inches drawn on my pad. 5 inches is > precise. You do?? How did you do that? Did you set your compass so that the points were exactly sqrt (5/pi) inches apart? What you probably have really is a circle that approximates 5 sq inches enough to suit YOU. Kyle Evans. _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Where's the flaw in my thinking?
Okay, I was sitting there the other day thinking about a non-FFT squaring algorithm... Say we have 14, which in binary is 1110... If we left shift this by the position of the 1, for each 1 in the binary representation, and add them together, we should get the square... So to square 14, we do this: 1110 << 3 == 111 + 1110 << 2 == 0111000 + 1110 << 1 == 0011100 + == 11000100 which is 196 So for each squaring, we have x left shifts and adds, where x is no larger that p. In any case... is this just me being dumb and missing that this is just a stupid way of squaring a number? _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: pi
> 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. A precisely defined value to me is a > finite value. > A never ending value is not finite. Ah, OK. Enlightenment dawns. You're using the words in a way which is completely different from the mainstream mathematical usage. While you're entirely free to do so, please don't be surprised if people misunderstand you. The conventional meaning of "infinite", at least in mathematical usage, is something like "greater than any specified quantity, no matter how large". A quantity such as e (2.718281828459045...), pi (3.14159265358979...) or 1/7 (0.142857142857...) may each require an infinite number of digits to represent their values *exactly* in the decimal representation, but they are most assuredly not infinite themselves. To see this, note that not one of them is greater than 4. Paul _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
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. A precisely defined value to me is a finite value. A never ending value is not finite. >Just as you can never get to the end of pi, though its value is known, you >can never PRECISELY note the area of a circle -- you can only express it >more and more accurately, depending on how accurate the value of PI you use is. I have a circle with a area of 5 square inches drawn on my pad. 5 inches is precise. Dan > _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Can I reserve the poaching thread? ;-)
On Fri, 11 Feb 2000, Paul Landon wrote: >May I exclusively reserve the topic of "poaching"? >;-) >I will release it in 60 days. (unless it is completed by >someone else before then). I think we should forge ahead. The hunter poached eggs and the blacksmith forged checks. phma _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Can I reserve the poaching thread? ;-)
May I exclusively reserve the topic of "poaching"? >;-) I will release it in 60 days. (unless it is completed by someone else before then). Paul Landon _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: "How Many Angels, How Many Pins?" Or PrimeNet's Assignment Strategy
On Thu, 10 Feb 2000, Stefan Struiker wrote: >Team M: > >Is it better to have 10 pins at 500MHz with one or two angels dancing, >or one 5GHz behemoth looking for luck, given the way PrimeNet >assigns exponents? Only ten pins?? Even a 64K microcontroller has forty! phma _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne Digest V1 #692
Mersenne Digest Friday, February 11 2000 Volume 01 : Number 692 -- Date: Thu, 10 Feb 2000 15:54:14 +0800 From: "Dave Mullen" <[EMAIL PROTECTED]> Subject: Mersenne: Base-3 Pseudoprimes This is a multi-part message in MIME format. - --=_NextPart_000_001E_01BF73DF.147B7640 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable I was wondering if base-3 pseudo-prime testing might be considerably = faster than LL testing for Mersenne Primes ? The base-3 pseudo-prime test is defined as :- 3 ^ P =3D=3D 3 (mod P) where P is a probable-prime (base-3 prp) 3 ^ P <> 3 (mod P) where P is composite We know that using binary exponentiation saves having to calculate every = power of 3, just some intermediates and the final required answer ... e.g. For M(3) =3D 7, (111 in binary), starting with the value 3, square, = multiply by 3, square, multiply by 3 This is effectively 4 multiplys to get our answer. If we used this modified form of the test :- 3 ^ (P+1) =3D=3D 9 (mod P) then we'd only need 3 multiplys i.e. starting with value 3, square, = square, square. This situation improves with higher exponents i.e. M(13) =3D 8191, = (1 in binary). Usual method takes 24 multiplys, modified method only takes 13 = multiplys. i.e.for a given Mersenne Exponent, usual method takes (Exp * 2) - 2 = multiplies, modified method takes (Exp) multiplys. Now, okay, this is still 2 more multiplys than the LL test would take. = But I believe there must be some advantages. 1) For the LL test using FFT multiplys, we have to Transform the data, = Multiply the matrices, Inverse Transform the data, then subtract 2. I am = assuming that the only reason we do the Forward and Inverse Transform = each time is so we can subtract the 2 ?=20 Please correct me here if I'm wrong ! (by the way, I would appreciate some pointers to FFTs for large integer = multiplication for idiots - I understand the (very) basic concept, but = how does the modulus occur automatically, why do we need to scramble the = data, why does it all work etc.) 2) If we are doing base-3 prps, the only operation we need is multiply - = so therefore we Transform the start value 3, then multiply the matrix by = itself as many times as required, then finally do one Inverse Transform. = This must save a lot of time over traditional LL testing, all those = Transforms jumping in and out of virtual memory (a lot of us don't have = 128+ MB of RAM !!). 3) The base-3 prp does not prove primality 100%, but it does prove = compositeness. So we might narrow down our range of potential Mersenne = exponents more quickly. And if if turns out that when we test the = remaining exponents, the LL test fails - what the hell, we just found a = new Carmichael Number - might still be a new record !! Dave - --=_NextPart_000_001E_01BF73DF.147B7640 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable I was = wondering if base-3=20 pseudo-prime testing might be considerably faster than LL testing for = Mersenne=20 Primes ? The base-3 pseudo-prime test is defined as = :- 3 ^ P =3D=3D 3 (mod P) where P is a probable-prime = (base-3=20 prp) 3 ^ P <> 3 (mod P) where P is = composite We know that using binary exponentiation saves = having to=20 calculate every power of 3, just some intermediates and the final = required=20 answer ... e.g. For M(3) =3D 7, (111 in binary), starting with = the value 3,=20 square, multiply by 3, square, multiply by 3 This is effectively 4 multiplys to get our=20 answer. If we used this modified form of the test = :- 3 ^ (P+1) =3D=3D 9 (mod P) then we'd only need 3 multiplys i.e. starting with = value 3,=20 square, square, square. This situation improves with higher exponents i.e. = M(13) =3D=20 8191, (1 in binary). Usual method takes 24 multiplys, modified method = only takes 13=20 multiplys. i.e.for a given Mersenne Exponent, usual method = takes (Exp *=20 2) - 2 multiplies, modified method takes (Exp) multiplys. Now, okay, this is still 2 more multiplys than the = LL test=20 would take. But I believe there must be some advantages. 1) For the LL test using FFT multiplys, we have to = Transform=20 the data, Multiply the matrices, Inverse Transform the data, then = subtract 2. I=20 am assuming that the only reason we do the Forward and Inverse = Transform=20 each time is so we can subtract the 2 ? Please correct me here if I'm wrong ! (by the way, I would appreciate some pointers to = FFTs for=20 large integer multiplication for idiots - I understand the (very) basic = concept,=20 but how does the modulus occur automatically, why do we need to scramble = the=20 data, why does it all work etc.) 2) If we are doing base-3 prps, the only operation = we need is=20 multiply - so therefore we Transform the start value 3, then multiply =
Mersenne: Re: Base e arithmetic
In a message dated 10/02/00 07:19:48 GMT Standard Time, [EMAIL PROTECTED] (Aaron Blosser) writes: << What I'm getting at is that at some point, pi reaches a practical limit at which expanding more decimal points is an abstraction because we could never measure anything large enough for it to be useful. I mean, c'mon! The universe is only so big! :-) >> This reminds me of a discussion I had years ago about the optimum design for a currency. Most countries have coins/notes of value 1, 2, 5, 10, 20, 50, 100 basic currency units, but in Romaina I found they used 1, 3, 10, 30, 100 ... multiples. Presumably the object is to make it such that over an average of many transactions, the minimum number of coins/notes changes hands. The optimum ratio between successive currency units (notes/coins) seems to be between 2 and 3, and a friend suggested that e (2.71828...) would be the best value, i.e. we have coins and notes valued at 1, e, e^2, e^3 ... cents or whatever. Maybe pi could be expressed exactly in such a system. After all, e^(pi*i) = -1. This led to a discussion as to whether or not it is possible to have a number system based on a non-integer base. Maybe the great minds of GIMPS can contribute to this. Cheers, George. _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Email to mersenne
> Anyone know why a email with subject "Re: more info on pi" I sent 24 hrs, > 2-9-00, ago has not appeared on the list? I sent it with a reply all. Then > today, 2-10-00, I forwarded another copy directly to [EMAIL PROTECTED] > > Another message I sent at the same time titled "Pi and Greek" 2-10-00 > showed up on the list right away. > > Where did the two copies of "Re: more info on pi", 2-9-00, go? You can try sending it to the list again, and cc'ing me at the same time. I can watch and figure out the cause of the problem. There are several possible explanations. The most likely is that you using words like "help" or "subscribe" in the first few lines of your message, and majordomo misidentifies it as a misdirected admin request. gordoni (list admin) _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Record p-1 factors
Dear All, Firstly, apologies to anyone receiving multiple copies of this message. The following are the only six 30+ digit factors found by Pollard's p-1 of which I am aware: (1) p34=8222057557067636644603420882415653 p-1=2^2.3.17.23.43.11657.506797.1632809.1692107.2496721 N=917^43-1 By: Steward 1999 (2) p34=7146831801094929757704917464134401 p-1=2^8.5^2.11^2.23^2.821.3371.39209.3394739.47358559 N=575th Fibonacci Number By: Silverman 1989 (3) p33=451990098872812878233073602931247 p-1=2.3.7^2.29.1423.55457.1907071.5110913.68921857 N=782^49-1 By: Steward 1999 (4) p32=49858990580788843054012690078841 p-1=2^3.5.13.19.977.1231.4643.74941.1045397.11535449 N=2^977-1 By: Brent (5) p31=4302886402340485071334706663243 p-1=2.7^2.139.401.129757.183361.488893.67720951 N=303^49-1 By: Steward 1999 (6) p30=174463386657191516033932614401 p-1=2^8.5^2.17.37.1627.5387.68111.152081.477361 N=2^740+1 By: Baillie Please let me know of any others. I will allow a couple of weeks for replies and then collate them into an appendix to my website. As you will note from positions 1, 3 and 5 this is not entirely a detached or altruistic exercise! Regards, Andy _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: Double-Checking
> Assume that a set of 400,000 exponents get single checked, > and double checked, and the error rate per check is 0.5%. > If the error occurrence is independent, that means about 4000 > will not match. Of these 4000 then the triple checks would > have errors in about 20, and require a quadruple check which > most likely would match one of the previous results. But, just to clarify, even if both a first time check and a double check both fail and come up with a bad residue for some reason, the partial residues from each run won't match, so they'd still be caught and flagged for a triple check. A triple check will match the first or the second run, or maybe neither requiring a 4th run. It all sounds pretty solid to me. :-) _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
