Re: Mersenne: pi
On Wed, 9 Feb 2000 [EMAIL PROTECTED] wrote: Hi, I have been considering the possible role pi might play in the progression of mersennes. It is generally accepted that the value of pi is a never ending series. But when I look at the circle, the formula for the area of a circle with a radius of 6 inches is: A=pi*r^2 = 3.1416 * (6)^2 = 113.0976. We did not, however, use the full and correct expansion of pi in the calculation. Pi has been figured out to over a billion (not sure of the exact figure) digits with no apparent end or pattern. But when I look at a circle I see a finite area within the circle with no means of growing or escape. Logic seems to indicate that pi would have to be a finite exact value since the area in the circle is finite. Yep you're actually right, pi has a finite exact value. The problem isn't that Pi isn't finite, it's less than 4 so it's finite. The problem isn't that it isn't exact. The problem is that it can't be represented exactly in decimals which mens that when we write the expansion, we'll always have to make do with an approximation to the exact value. So, either the figure for pi is in error (not likely) or pi has a end. Any decimal representation of pi is in error, since it can only be an approximation. The end. What say ye? Dan I think where your argument slips is in confusing the number for it's representation, ie. how it's written. These are two different concepts, and confusing them leads to argumenting from false analogies. -- Henrik Olsen, Dawn Solutions I/S URL=http://www.iaeste.dk/~henrik/ Linux isn't at war. War involves large numbers of people making losing decisions that harm each other in a vain attempt to lose last. Linux is about winning.Alan Cox on linux-kernel _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Finite, Amicable, Pi...lots of topics
In Unsolved Problems in Number Theory, Richard K Guy says of Mersenne primes: "their number is undoubtedly infinite, but proof is hopelessly beyond reach". He then offers some suggestions for the size of M(x), the number of primes p = x for which 2^p -1 is prime. Gillies suggested M(x) ~ c ln x Pomerance suggested M(x) ~ c ( ln ln x ) ^ 2 This is very serious indeed, especially for those of us who believe the number of Mersenne primes to be finite. It's a fairly old book in a manner of speaking: in 1981 he poses the question as to whether 2^p - 1 is always square-free. I'm sure this has been discussed here from time to time - did we ever get an answer? In this case, Guy believes that the answer is no, and that it could be settled by computer if you were lucky. I'm still on topic if I talk about perfect numbers, where the sum of the factors of a perfect number n, which I call s(n) is equal to n. However I'm off topic as soon as I start talking about amicable numbers, sometimes called semi-perfect numbers. For a pair of amicable numbers m and n we have s(m) = n and s(n) = m. For example s(220) = 284 while s(284) = 220. I choose to mention these because of the recent mention of hairy and smooth numbers and in the context of Esau and Jacob, also recent players here, as the number 220 is of some significance in their story in Genesis 32:14. The recent heroes in this field are H J J te Riele, who "knows everything about amicable numbers" according to a now forgotten usenet poster and Lee and Madachy, who published "The history and discovery of Amicable Numbers" in the Journal of Recreational Mathematics in 1972, along with an alarmingly long list of then known amicable numbers. (Does anybody know if this journal is still published? When I subscribed to it for a while, though, it wasn't too recreational, and seemed obsessed with repunits for a while.) There are far more amicable pairs known than even perfect numbers, yet Guy's claim on their infinite number or otherwise is, surprisingly, weaker. "It is not known if there are infinitely many, but it is believed that there are." Finally, pi. Along with others, I have been amused by the reputed Alabama legislature decision, and spend a lot of time looking at the urban legends at http://www.snopes.com/ which is one of the most significant sites on the web, possibly second only to http://www.mersenne.org/prime.htm ? However, as I believe in the inerrancy of scripture, I obviously have a problem with 1 Kings 7:23. I don't believe either that pi = 3 or that God thinks pi = 3. So, what happens? At http://www.khouse.org/articles/biblestudy/19980401-158.html we can learn that there is a subtle difference in the text from what might be expected in that the word for circumference "qav" has been replaced by the word "qaveh". If we take note of the numerical values associated with these words, qav = 100 + 6, while qaveh = 100 + 6 + 5. Accordingly, we take the implied multiplicand of 3 and extend it by 111/106, which gives an approximation of 333/106, which is 3.141509... which is accurate enough for practical purposes. Possibly not for rocket science, but that's not what we're talking about here. K House probably don't phrase their explanation in the way I would choose, but it nevertheless makes compelling reading from a reasonably mainstream source. Over history, there have been numerous other approximations to the value of pi. Our current culture seems to favour 22/7 as an approximation, and the Biblical approximation above suggests 333/106. However, this is not the best available in three digits, which is, so far as I know, 355/113, which is correct to an astonishing one part in ten million. I understand that in certain quarters, 3 1/7 was not in vogue, with 3 1/8 favoured. What, argued these particular mystics, could be a better number than five squared shared by two cubed? N P Smith asked whether we should be more concerned by those who serious propose answers which are clearly wrong or by those who spend time in repeatedly refuting these spurious claims. As for squaring the circle, another popular pastime, the Greeks noted that a square of side 8 have pretty much the same area. This points to 256/81 or sixteen squared shared by nine squared if you like that sort of thing. It's still not exact. That's what irrational means... I'm sorry to have strayed off topic: at the moment I can't find any legitimate connection between pi and Mersenne numbers - if anybody can do so then obviously I am absolved because this posting will have been on topic after all. I am absolved! Between researching this article and posting it, others have started to explore the possibility of such links. Regards, Ian W Halliday Wellington, New Zealand --- Happiness is just around the corner. - D H Lehmer _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ --
RE: Mersenne: pi
The problem isn't that Pi isn't finite, it's less than 4 so it's finite. The problem isn't that it isn't exact. The problem is that it can't be represented exactly in decimals which mens that when we write the expansion, we'll always have to make do with an approximation to the exact value. Consider this: Let's assume that the universe is spherical (a logical assumption if we assume it's the result of a currently expanding explosion xx years ago). If we were to calculate the radius of this sphere down to a single atomic width, using some decently expanded version of pi would could come up with an exact number for the volume of the universe. 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! :-) Being in a hurry, I don't have the time to figure out how many decimal places that would be...perhaps someone more adventurous would care to give it a go. Aaron _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
At 12:06 AM 2/9/00 -0600, [EMAIL PROTECTED] wrote: But when I look at a circle I see a finite area within the circle with no means of growing or escape. Logic seems to indicate that pi would have to be a finite exact value since the area in the circle is finite. No, pi is irrational, which means that the digits go on forever without repeating. So, either the figure for pi is in error (not likely) or pi has a end. The calculated value of pi is never exact, since it is calculated to a finite precision. ++ | Jud McCranie | || | 137*2^197783+1 is prime! (59,541 digits, 11/11/99)| | 137*2^224879+1 is prime! (67,687 digits, 1/00)| ++ _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
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. 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. Thus, the limit of the area of a circle as your approximation for pi approaches an infinite expansion is pi*r^2. At 12:06 AM 2/9/00 -0600, you wrote: Hi, I have been considering the possible role pi might play in the progression of mersennes. It is generally accepted that the value of pi is a never ending series. But when I look at the circle, the formula for the area of a circle with a radius of 6 inches is: A=pi*r^2 = 3.1416 * (6)^2 = 113.0976. We did not, however, use the full and correct expansion of pi in the calculation. Pi has been figured out to over a billion (not sure of the exact figure) digits with no apparent end or pattern. But when I look at a circle I see a finite area within the circle with no means of growing or escape. Logic seems to indicate that pi would have to be a finite exact value since the area in the circle is finite. So, either the figure for pi is in error (not likely) or pi has a end. The end. What say ye? Dan _ 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
RE: Mersenne: pi
I am not a math man but I follow this discussion. Can we say the same to the case of 10 divide by 3 , The result is 3 1/3 but as a decimal way of writing you never end. Maybe I just miss the whole thing, in that case I am sorry, and will continue just reading this kind of topics. Bye, Paul van Grieken -Original Message- From: Jud McCranie [SMTP:[EMAIL PROTECTED]] Sent: Wednesday, February 09, 2000 3:22 PM To: [EMAIL PROTECTED] Cc: [EMAIL PROTECTED]; [EMAIL PROTECTED] Subject: Re: Mersenne: pi At 12:06 AM 2/9/00 -0600, [EMAIL PROTECTED] wrote: But when I look at a circle I see a finite area within the circle with no means of growing or escape. Logic seems to indicate that pi would have to be a finite exact value since the area in the circle is finite. No, pi is irrational, which means that the digits go on forever without repeating. So, either the figure for pi is in error (not likely) or pi has a end. The calculated value of pi is never exact, since it is calculated to a finite precision. ++ | Jud McCranie | || | 137*2^197783+1 is prime! (59,541 digits, 11/11/99)| | 137*2^224879+1 is prime! (67,687 digits, 1/00)| ++ _ 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
RE: Mersenne: pi
Actually bro, I highly doubt the Universe is spherical. If one is to believe in the concept of gravity bending space-time, then the Universe would be more akin to some bubbly, blobby, amorphous structure (see http://www.sciam.com/1999/0699issue/0699landy.html) Interestingly enough, you could conceptually have "bubbles" in space-time which are cut off from the rest of the universe. Anyway, back to Pi... I think that the major issue is that we look at a circle and think of some number of units, atoms, particles, whatever... However, we neglect the fact that a circle isn't constituted of particles... This seems to been some weird human characteristic, we think of everything in units (time for example)... Take the function y=x, there are an infinite number of points on that line... Even, if I limit the range of x from (-1,1), there are still an infinite number of points on that line... So even the simple function y=x has infinite precision, yet I can precisely determine that the length of that line is 2*sqrt(2). -Original Message- From: Aaron Blosser [mailto:[EMAIL PROTECTED]] Sent: Wednesday, February 09, 2000 8:41 AM To: Mersenne@Base. Com Subject: RE: Mersenne: pi The problem isn't that Pi isn't finite, it's less than 4 so it's finite. The problem isn't that it isn't exact. The problem is that it can't be represented exactly in decimals which mens that when we write the expansion, we'll always have to make do with an approximation to the exact value. Consider this: Let's assume that the universe is spherical (a logical assumption if we assume it's the result of a currently expanding explosion xx years ago). If we were to calculate the radius of this sphere down to a single atomic width, using some decently expanded version of pi would could come up with an exact number for the volume of the universe. 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! :-) Being in a hurry, I don't have the time to figure out how many decimal places that would be...perhaps someone more adventurous would care to give it a go. Aaron _ 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: Re: Mersenne : pi
Date: Wed, 09 Feb 2000 10:50:44 -0500 From: Jeff Woods [EMAIL PROTECTED] Subject: Re: Mersenne: pi 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. 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. Actually what you're saying is, you can PRECISELY know the area of a circle, or PRECISELY know the diameter of a circle, but not both, without resorting to using the symbol pi. Sounds like an instance of the Pisenberg Uncertainty Principle to me Well, back to lurking. Phil Brady _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Re: optimizing for Athlon
Brian Beesley wrote: The optimization that should probably be done for Athlon is to organize the code to allow FMUL FADD to execute in parallel (which the Pentium II/III core just can't manage). This could give a speedup of the order of 40%. That would be nice if true, but I suspect it's a bit overoptimistic. The reason is this: the Athlon utilizes out-of-order execution, i.e. even if the assembly code indicates a certain instruction ordering (e.g. FADDs interleaved with FMULs, as required for the Pentium, which can complete just one double-precision floating op per cycle), the CPU is free to execute them in a different order, as long as any data dependencies are preserved. That means the Athlon is probably already executing quite a few such FADD/FMUL pairs in parallel, unless I'm misunstanding something fundamental about its OOE capabilities. As I've found the available Athlon documentation (the technical brief and the code optimization guide from the AMD website) to be frustratingly vague about things like the register set architecture and the functional units, can anyone answer the following for me? 1a,b,c) How many floating-point registers does the Athlon have? Are these all 80 bits? Are they accessed via the same kind of stack-based model as the Pentium? 2a,b,c) I believe the Athlon has two floating adders in addition to a floating multiplier. Can it dispatch 2 FADDs and 1 FMUL per cycle? Can it do 2 double- precision FADDs per cycle, or just do single-precision adds in parallel? (The former would help with the higher-radix FFTs in an LL code, since these have more adds than multiplies, but the latter, while nice for multimedia applications, would be useless for speeding LL testing.) Thanks, -Ernst _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
Can we say the same to the case of 10 divide by 3 , The result is 3 1/3 but as a decimal way of writing you never end. Almost, but the difference is 10 / 3 is expressable as one number divided by another. to my kowledge there is no a and b where a/b=pi. Someone correct me if I am wrong. -Original Message- From: Grieken, Paul van [EMAIL PROTECTED] To: 'Jud McCranie' [EMAIL PROTECTED] Cc: [EMAIL PROTECTED] [EMAIL PROTECTED] Date: Wednesday, February 09, 2000 11:53 AM Subject: RE: Mersenne: pi I am not a math man but I follow this discussion. Can we say the same to the case of 10 divide by 3 , The result is 3 1/3 but as a decimal way of writing you never end. Maybe I just miss the whole thing, in that case I am sorry, and will continue just reading this kind of topics. Bye, Paul van Grieken -Original Message- From: Jud McCranie [SMTP:[EMAIL PROTECTED]] Sent: Wednesday, February 09, 2000 3:22 PM To: [EMAIL PROTECTED] Cc: [EMAIL PROTECTED]; [EMAIL PROTECTED] Subject: Re: Mersenne: pi At 12:06 AM 2/9/00 -0600, [EMAIL PROTECTED] wrote: But when I look at a circle I see a finite area within the circle with no means of growing or escape. Logic seems to indicate that pi would have to be a finite exact value since the area in the circle is finite. No, pi is irrational, which means that the digits go on forever without repeating. So, either the figure for pi is in error (not likely) or pi has a end. The calculated value of pi is never exact, since it is calculated to a finite precision. ++ | Jud McCranie | || | 137*2^197783+1 is prime! (59,541 digits, 11/11/99)| | 137*2^224879+1 is prime! (67,687 digits, 1/00)| ++ _ 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 _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
"Aaron" == Aaron Blosser [EMAIL PROTECTED] wrote the following on Wed, 9 Feb 2000 07:40:37 -0700 Aaron If we were to calculate the radius of this sphere down to a Aaron single atomic width, using some decently expanded version of Aaron pi would could come up with an exact number for the volume of Aaron the universe. Aaron What I'm getting at is that at some point, pi reaches a Aaron practical limit at which expanding more decimal points is an Aaron abstraction because we could never measure anything large Aaron enough for it to be useful. I mean, c'mon! The universe is Aaron only so big! :-) Aaron Being in a hurry, I don't have the time to figure out how Aaron many decimal places that would be...perhaps someone more Aaron adventurous would care to give it a go. Cecil Adams mentioned this in one of his columns a few years ago; you can find a copy online at: http://www.straightdope.com/classics/a3_357.html Cecil's column is intended for the general public, not those trained in mathematics, so it's a good "everybody can understand" discussion. Executive summary: Pi to 35 decimal places lets you compute the circumference of the universe with an error of less than the radius of a hydrogen atom. Pi to 39 decimal places lets you compute the circumference of the universe with an error of less than the radius of the NUCLEUS of a hydrogen atom. Mike _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: Re: Mersenne Digest V1 #690
What I would love to see on the individual account report is the work rate of each machine.. eg. P90 hrs/day total and P90 hrs/day in the last month or so. Is this possible Lawrence.. In a message dated 09/02/70, 10:26:53, [EMAIL PROTECTED] writes: Would it be possible to add a CPU type/speed column to the "Exponents Assigned" list of the "Individual Account Report"? I second that, and move that a program/version line also be added, to both the individual account report and the cleared exponents list. _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: PrimeNet Top Producers List
From: Scott Kurowski Following the database synchro we performed on PrimeNet yesterday... I noticed the addition of the machine speeds and software version to my individual account report. Now that's what I call being responsive to user requests! 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: PrimeNet Top Producers List
Scott wrote:- Following the database synchro we performed on PrimeNet yesterday, we cleaned up some of the 'dead' user accounts over a year old. The cumulative machine times were added to the Entropia.com, Inc. 'challenge' account, the first one opened on PrimeNet in April 1997. Hopefully nobody will mind our reclaiming the fragmented time. :-) Do people using the manual check out forms get in the Top Producers list? I ask because I've never been able to find myself in the list and I had an email from a former GIMPS contributor who claimed he got no credit for exponents tested through the manual check out/check in pages. Bill Rea, Information Technology Services, University of Canterbury \_ E-Mail b dot rea at its dot canterbury dot ac dot nz / New Phone 64-3-364-2331, Fax 64-3-364-2332/) Zealand Unix Systems Administrator (/' _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Re: optimizing for Athlon
On Wed, 9 Feb 2000 [EMAIL PROTECTED] wrote: As I've found the available Athlon documentation (the technical brief and the code optimization guide from the AMD website) to be frustratingly vague about things like the register set architecture and the functional units, can anyone answer the following for me? That's putting it kindly. AMD's Athlon optimization manual sucks bigtime. They forget to list the actual freakin' latencies of *any* instructions, but remember to point out the "industry-leading" and "industry-standard" features of the chip. I hope I never see these two terms in a technical manual again. 1a,b,c) How many floating-point registers does the Athlon have? Are these all 80 bits? Are they accessed via the same kind of stack-based model as the Pentium? Apparently the Athlon has three floating point pipelines: 1 FMUL, 1 FADD, and 1 store pipeline. These split 3DNow and MMX instructions between them as well as the FPU ones; the only other FPU data we are given is that there are 88 floating point registers in the register file. No examples, no stall rules, no latencies, no store bypassing rules, no decoding rules (for FP) nothing. AMD's K6 family optimization manual was exactly the same. Lots of luck getting any performance tuning done at all. It's not even clear that FPU instructions can issue in parallel with integer instructions (I believe they can, but the two units share the same three decoders). I would love for someone to club me over the head and tell me where all that info is located jasonp _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Re: optimizing for Athlon
On 9 Feb 00, at 15:36, [EMAIL PROTECTED] wrote: The optimization that should probably be done for Athlon is to organize the code to allow FMUL FADD to execute in parallel (which the Pentium II/III core just can't manage). This could give a speedup of the order of 40%. That would be nice if true, but I suspect it's a bit overoptimistic. The reason is this: the Athlon utilizes out-of-order execution, i.e. Yes, that's why I wrote "could", as opposed to "should". How much benefit you get from OOE depends to an enormous extent on how the code is organized. If you've just retired registers containing temporary results which you need back to work on _right now_ then you could be working rather inefficiently. You've made the point in the past that organizing HLL source code "properly" gives the optimizer in the compiler a better chance of doing a decent job; the same is no less true in that well-organized assembler code gives the execution scheduler in the CPU less of a chance to foul things up. 1a,b,c) How many floating-point registers does the Athlon have? Are these all 80 bits? Are they accessed via the same kind of stack-based model as the Pentium? From the briefing notes I have (which are quite elderly and may not correspond with the consumer silicon); so far as x86 compatible FP operations are concerned, a) there are 40 FPU registers but only 8 of them are named. (The others are available to hold temporaries etc). This register pool is shared with the 3D-Now instruction set. b) Yes. (In 3D-Now mode they actually contain 128 bits) c) The 8 named FP registers are logically organized as a stack just like the Intel model. (Unchanged since the 8087!) 2a,b,c) I believe the Athlon has two floating adders in addition to a floating multiplier. Can it dispatch 2 FADDs and 1 FMUL per cycle? Can it do 2 double- precision FADDs per cycle, or just do single-precision adds in parallel? a) There are two independent 80-bit FP execution units, both can do FADD but only one can do FMUL. b) No. You can do 2 FADDs or 1 FADD + 1 FMUL per cycle. c) I think in 3D-Now mode you can do 4 SP operations in parallel in each execution unit instead of one 80-bit operation. Regards Brian Beesley _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
At 02:31 AM 2/9/00 -0500, gav wrote: I think my favorite counterexample to arguments like this is Gabriel's Horn. Take the function 1/x, and revolve it around the x-axis. You now have something that looks very similar to a trumpet's bell. Now, find the volume of this from 0 to infinity. It has a finite volume. However, it has an infinite surface area. I have a little trouble conceptualizing what would happen if you fill this horn with paint. If you completely fill this horn with paint (a finite volume), the inner surface of the horn should be completely covered with paint, right? But the inner surface of the horn has infinite area, so wouldn't it take an infinite amount of paint to paint it? Where is my intuition going wrong? Mike _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: pi
-Original Message- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]]On Behalf Of Mike Bandsmer Sent: Wednesday, February 09, 2000 7:32 PM To: [EMAIL PROTECTED] Subject: Re: Mersenne: pi At 02:31 AM 2/9/00 -0500, gav wrote: I think my favorite counterexample to arguments like this is Gabriel's Horn. Take the function 1/x, and revolve it around the x-axis. You now have something that looks very similar to a trumpet's bell. Now, find the volume of this from 0 to infinity. It has a finite volume. However, it has an infinite surface area. I have a little trouble conceptualizing what would happen if you fill this horn with paint. If you completely fill this horn with paint (a finite volume), the inner surface of the horn should be completely covered with paint, right? But the inner surface of the horn has infinite area, so wouldn't it take an infinite amount of paint to paint it? Where is my intuition going wrong? It would take infinite area of an infinitesimally thin layer of paint, which would have no volume due to its thinness. Since paint can't be infinitely thin, this also means you can't actually fill the object with paint, because there will be volume in areas into which paint molecules can't fit. Mike _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: PI is a transcendental number
I recall my study of Maths in high school:- 1. First we learn about Integers : 0, 1, 2, 3,.. positive and negative 2. Then about Decimals : 0.1, 0.23, 3.5 etc 3. Follow by Fractions in the form of a/b where a and b are integers. 4. By converting fractions to decimals, we discover infinite but repetitive sequence e.g. 2/9 = 0.; 17/27 = 0.629629629...with infinite repetition of 629 5. And the study of geometry and algebra introduce Irrational numbers. e.g. square root of 2 = 1.414213562373 to infinity small without any repetitive sequence. Basically, an irrational number is one that cannot be expressed by a fraction of integers. And any numbers that can be expressed by a fraction is called rational number. 6. But most irrational numbers can be obtained from solving a polynomial equations e.g. x**2-2 = 0 gives rise to x = +/- sqrt root 2. 7. And we learn about Imaginary number from solving equation such as X** 2 + 4 = 0 gives rise to x = +/- 2i 8. Finally, Pi and "e" were introduced as Transcendental numbers :- Those irrational numbers that cannot be derived from the roots of any Polynomial Equations! Integers, Decimals, Fractions, Irrational, and even Transcendental Numbers, they are all FINITE and PRECISE (can be precisely defined). Such properties hold irregard what sort of Numeric Representation (such as Binary, Hexadecimal etc). 9. Euler had marvellously combined all the above into one equation "e" to the power of i (imaginary) * PI = -1. Thanks to my Maths teachers for showing the wonders of the Numbering System HweeBoon _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
You're on the right track, but the mistake you're making is that the paint can be infinitesimally thin in order to coat the surface. So, if the thickness of the paint decreases proportionately to the function, then you've only used a finite amount of paint (as the volume is only finite), but you've coated an infinite surface area. If someone happens to remember the exact way the integral are written, that'd be a big help. I'm going to try and find my old Calc text now, I'm sure it's in there somewhere. gav At 04:32 PM 2/9/00 -0800, Mike Bandsmer wrote: At 02:31 AM 2/9/00 -0500, gav wrote: I think my favorite counterexample to arguments like this is Gabriel's Horn. Take the function 1/x, and revolve it around the x-axis. You now have something that looks very similar to a trumpet's bell. Now, find the volume of this from 0 to infinity. It has a finite volume. However, it has an infinite surface area. I have a little trouble conceptualizing what would happen if you fill this horn with paint. If you completely fill this horn with paint (a finite volume), the inner surface of the horn should be completely covered with paint, right? But the inner surface of the horn has infinite area, so wouldn't it take an infinite amount of paint to paint it? Where is my intuition going wrong? Mike _ 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
Re: Mersenne: PI is a transcendental number
At 09:58 AM 2/10/00 +0800, Low Hwee Boon wrote: 6. But most irrational numbers can be obtained from solving a polynomial equations Actually almost all irrational numbers are transcendental, and therefore not the root of a polynomial with rational coefficients. ++ | Jud McCranie | || | 137*2^197783+1 is prime! (59,541 digits, 11/11/99)| | 137*2^224879+1 is prime! (67,687 digits, 1/00)| ++ _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi, limits, and other things OT
Hi folks
I think my favorite counterexample to arguments like this is Gabriel's
Horn. Take the function 1/x, and revolve it around the x-axis. You now
have something that looks very similar to a trumpet's bell. Now, find
the
volume of this from 0 to infinity. It has a finite volume. However, it
has an infinite surface area.
If someone happens to remember the exact way the integral are written,
that'd be a big help. I'm going to try and find my old Calc text now, I'm
sure it's in there somewhere.
If y=f(x), the volume of revolution is given by
{integral from 1 to infinity) pi.y^2 dx
Note the integral starts at 1, not zero (otherwise the volume is undefined)
for the Horn. The volume is in fact pi.
The surface of revolution is given by
(integral from 1 to infinity) 2.pi.y.sqrt(1+y'^2) dx
where y'=dy/dx= -1/x^2 in the case of the horn.
The integrand is 2.pi.x /sqrt(x^4+1). If you recognise this, good for you
(Apply a change of variable t=x^2 and you will get pi. 1/sqrt(t^2+1) under
the integral, which you might recognize. If you're still stuck, think about
arcsinh t).
Recognize it or not, it really doesn't matter, Note the integrand is
actually a little greater than
2 pi y dx
which is the usual mistake first made with surfaces of revolution
(approximating the surface by 'delta-x height cylinders' instead of 'delta-x
height slices of cones'). However this function is a lot easier to recognize
as the derivative of 2.pi.ln x, and so the integral as we approach infinity
is indeed unbounded.
I have a little trouble conceptualizing what would happen if you fill
this
horn with paint. If you completely fill this horn with paint (a finite
volume), the inner surface of the horn should be completely covered with
paint, right? But the inner surface of the horn has infinite area, so
wouldn't it take an infinite amount of paint to paint it? Where is my
intuition going wrong?
It's a bit like the old gag "how many lawyers does it take to wallpaper a
room?" ("Depends how thinly you slice them"). A given amount of paint or
lawyers can cover an arbitrarily large surface provided you spread it thinly
enough. Not possible in the real world of course (not to mention the Horn's
neck is ultimately too narrow to squeeze a paint molecule down), but
mathematicians aren't limited by such physical constraints.
Single-celled organisms have known for eons that the best way to improve
their rate of nutrition is to stretch their volume into the largest possible
surface area. Fortunately physics intervenes and an infinitesimally thin
organism of infinite length but finite volume isn't a biological
possibility.
Chris Nash
Lexington KY
UNITED STATES
_
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Mersenne: Re: Mersenne Digest V1 #690
It has a finite volume. However, it has an infinite surface area. One of my friends, AYL (who proofread my Mersennes paper) loves to talk about Gabriel's Horn. His favorite comment is: "So, I can pour paint INTO the thing, but I can't paint it?" Stephan Lavavej _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: pi
At 08:35 PM 2/9/2000 -0500, [EMAIL PROTECTED] wrote: It would take infinite area of an infinitesimally thin layer of paint, which would have no volume due to its thinness. Since paint can't be infinitely thin, this also means you can't actually fill the object with paint, because there will be volume in areas into which paint molecules can't fit. Mike Filling the horn with paint has a couple additional problems: since it is an infinitely long capillary, filling time would be infinity^4 or so (laminar flow conductance being proportional to diameter^4 and inversely proportional to length) A realizable section of Gabriel's horn would necessarily be lumpy when constructed of real material. Think of a tube constructed of soccer balls glued together. If the horn inner diameter is a kilometer, great, it looks pretty smooth. (Say for the sake of argument the diameter of these soccer balls is 3 decimeters.) But further along, where the inner diameter has fallen off to one meter, it's beginning to look pretty lumpy already, and when inner diameter drops to 1 decimeter, the tube roughness is very significant. Now move out to where the inner diameter is 1 Angstrom, and the atoms of which the wall is constructed are 3 Angstroms diameter, and it looks the same. I'm surprised noone responded about continued fractions to Ian Halliday: At 10:42 PM 2/9/2000 +1300, [EMAIL PROTECTED] wrote: Over history, there have been numerous other approximations to the value of pi. Our current culture seems to favour 22/7 as an approximation, and the Biblical approximation above suggests 333/106. However, this is not the best available in three digits, which is, so far as I know, 355/113, which is correct to an astonishing one part in ten million. I understand that in certain quarters, 3 1/7 was not in vogue, with 3 1/8 favoured. What, argued these particular mystics, could be a better number than five squared shared by two cubed? N P Smith asked whether we should be more concerned by those who serious propose answers which are clearly wrong or by those who spend time in repeatedly refuting these spurious claims. PI~=3.1415926535897932384626433832795 subtract the integer part, take the reciprocal of the rest, and iterate, to produce the continued fraction's coefficients. Reassemble successively increasing numbers of terms, until the rational number obtained is sufficiently accurate. This is an effective method of determining gear ratios approximating arbitrary reals. 3+ 1 / (7 + 1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/(3+1/(1+1/(14+1/(2+1/(1+... ))) 3= 3 4= 3 +1 3.14 2857142857... =3+1/7 = 22/7 3.1 25 = 3+1/(7+1) = 25/8 3.1415 09433962264150943396226415... =3+1/(7+1/15) = 3 + 15/106 = 333/106 3+1/(7+1/(15+1/)) =355/113, see below 3.141592 9203539823008849557522124... =3+1/(7+1/(15+1/1)) = 3 + 1/(7+1/16)= 3+1/(113/16) = 3+ 16/113 = 355/113 3+1/(7+1/(15+1/(1+1)))= 3.1415 525114155251141552511415525 3+1/(7+1/(15+1/(1+1/292))) = 103993/33102 = 3.141592653 0119026040722614947737 3+1/(7+1/(15+1/(1+1/293)))= 3.141592653 9214210447087159415927 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/1) = 3.141592653 4674367055204547853492 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1)) = 3.141592653 6189366233975003014106 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1))) = 3.1415926535 583573009183052053374 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/2))) = 3.14159265358 1012044193065819 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1 = 3.1415926535 914039784825424142193 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1)= 3.14159265358 70561991705458087813 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/3)=3.14159265358 9 3891715436873217069 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/(3+1))=3.1415926 53589 8153832419437773074 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/(3+1/2))=3.14159 2653589 6274836288508219852 3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/(3+1/(1+1/14)))= 80143857/25510582=3.14159265358979 26593756269457122 Ken Ken Kriesel, PE [EMAIL PROTECTED] _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne Digest V1 #690
Mersenne Digest Wednesday, February 9 2000 Volume 01 : Number 690 -- Date: Tue, 8 Feb 2000 16:18:30 - From: "Brian J. Beesley" [EMAIL PROTECTED] Subject: Re: Mersenne: Hypothesis On 7 Feb 00, at 22:40, Wojciech Florek wrote: Hi all! Due to some reasons I've considered numbers in a form 3*2^n (3,6,12,) and I've found that almost in each interval 3*2^n..3*2^(n+1) there are one, two or three exponents of Mersenne prime. Isn't this really just saying that Mersenne primes have a similar distribution to the series k*2^n? I thought we already had a hypothesis suggesting that there should be about 1.4 Mersenne primes per octave - on average - which is actually a slightly more informative version of the hypothesis based on the observation reported here. The first two: 2,3 are below or equal 3*2^0. `Almost' means that there is a true gap for n=6: there are no exponents between 3*64=192 and 3*128=384. So the hypothesis has a counter-example ... Sorry, I'll try to be more constructive. If you have a hypothesis that there are, on average, k Mersenne primes per octave but that the distribution is random, if you sample the number of Mersenne primes per octave (starting at _any_ point) then you should get something like a Poisson distribution with mean k. It might be, from the limited sample we have, starting the sampling interval at 3*2^n (as opposed to q*2^n for some other q) gives a better/smoother fit than others - I don't know, I haven't tried - but, in any case, the sample size we have to go on is pitifully small for testing the hypothesis. The other possible gap is for n=20 3*2^20=3145728..6291456, but this is a reminder of the v17 bug (???). We haven't fully searched this interval yet, and double-checking is nowhere near complete. I'd guess that the bad v17 results have been redone long since. Among 36 considered exponents (without 2,3) 25 can be written as 3*2^n+p OR (sometmes AND) 3*2^n-p, where p is a prime. On the other hand, 11 exponents are expressed as 3*2^n +/- c, where c is a composite number. I've considered only differences with interval limits. The smallest is 2203=3*512+(5*149)=3*1024-(11*79). The others are: 2281,11213,44497,86243,110503,132049,216091, 756839,859433,1257787,2976221. Note that the two largest known exponents are 3021377=3*2^20-124351 [prime!] 6972593=3*2^21+681137 [prime!] This is an interesting observation. Do we have a handle on how likely it is that an arbitary number of a similar size can be represented this way? I have a (probably incorrect, gut) feeling that the "composites" are under-represented. However, even if we found a relationship like this which is true for all known Mersenne primes, we wouldn't be sensible to use it as a criterion for eliminating exponents without a decent proof (not just a hand-waving argument) as to why the relationship _must_ hold for Mersenne prime exponents. Regards Brian Beesley _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers -- Date: Tue, 8 Feb 2000 17:57:03 +0100 From: "Hoogendoorn, Sander" [EMAIL PROTECTED] Subject: RE: Mersenne: Request for feature As long as we're asking, is it possible to slit up the Assignments Report and the Cleared Exponents Report into seperate lists for double checking,first time checking and factoring, these lists get to large to download often. - -Original Message- From: Alan Vidmar [mailto:[EMAIL PROTECTED]] Sent: maandag 7 februari 2000 18:26 To: [EMAIL PROTECTED] Subject: Mersenne: Request for feature Scott, Would it be possible to add a CPU type/speed column to the "Exponents Assigned" list of the "Individual Account Report"? This info seems to be collected as the "Machines Assigned to PrimeNet" suggests. Thanks, Alan _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers -- Date: Tue, 8 Feb 2000 11:50:45 -0500 From: "Conor McCutcheon" [EMAIL PROTECTED] Subject: Re: Mersenne: AMD Athlon problems Well, I wish it was as simple as power management, but turning that off in the bios and windows is the very first thing I did after installing the CPU. As for the CPU overheating, I really doubt it, but it is worth looking into. I have what appears to be a good fan attached to the CPU, one designed to be used the way I am using, with the heat transfer goop on it as well, and I have an additional case fan that brings in cool air from the room. All appear to be working. As for the suggestion that the CPU speed is incorrectly set, it is not. I have checked that setting and played with it enough to be sure
Mersenne Digest V1 #691
Mersenne Digest Wednesday, February 9 2000 Volume 01 : Number 691 -- Date: Wed, 9 Feb 2000 10:28:25 -0600 From: Jeremy Blosser [EMAIL PROTECTED] Subject: RE: Mersenne: pi Actually bro, I highly doubt the Universe is spherical. If one is to believe in the concept of gravity bending space-time, then the Universe would be more akin to some bubbly, blobby, amorphous structure (see http://www.sciam.com/1999/0699issue/0699landy.html) Interestingly enough, you could conceptually have "bubbles" in space-time which are cut off from the rest of the universe. Anyway, back to Pi... I think that the major issue is that we look at a circle and think of some number of units, atoms, particles, whatever... However, we neglect the fact that a circle isn't constituted of particles... This seems to been some weird human characteristic, we think of everything in units (time for example)... Take the function y=x, there are an infinite number of points on that line... Even, if I limit the range of x from (-1,1), there are still an infinite number of points on that line... So even the simple function y=x has infinite precision, yet I can precisely determine that the length of that line is 2*sqrt(2). - -Original Message- From: Aaron Blosser [mailto:[EMAIL PROTECTED]] Sent: Wednesday, February 09, 2000 8:41 AM To: Mersenne@Base. Com Subject: RE: Mersenne: pi The problem isn't that Pi isn't finite, it's less than 4 so it's finite. The problem isn't that it isn't exact. The problem is that it can't be represented exactly in decimals which mens that when we write the expansion, we'll always have to make do with an approximation to the exact value. Consider this: Let's assume that the universe is spherical (a logical assumption if we assume it's the result of a currently expanding explosion xx years ago). If we were to calculate the radius of this sphere down to a single atomic width, using some decently expanded version of pi would could come up with an exact number for the volume of the universe. 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! :-) Being in a hurry, I don't have the time to figure out how many decimal places that would be...perhaps someone more adventurous would care to give it a go. Aaron _ 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 -- Date: Wed, 9 Feb 2000 12:00:26 EST From: [EMAIL PROTECTED] Subject: Mersenne: Re: Mersenne : pi Date: Wed, 09 Feb 2000 10:50:44 -0500 From: Jeff Woods [EMAIL PROTECTED] Subject: Re: Mersenne: pi 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. 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. Actually what you're saying is, you can PRECISELY know the area of a circle, or PRECISELY know the diameter of a circle, but not both, without resorting to using the symbol pi. Sounds like an instance of the Pisenberg Uncertainty Principle to me Well, back to lurking. Phil Brady _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers -- Date: Wed, 09 Feb 2000 14:45:44 -0500 From: Bassam Abdul-Baki [EMAIL PROTECTED] Subject: Re: Mersenne: Re: Mersenne : pi Actually, you can express PI in heaxadecimal form. This was proven by Simon Plouffe. A decimal expression is still unknown. Bassam Abdul-Baki [EMAIL PROTECTED] wrote: Date: Wed, 09 Feb 2000 10:50:44 -0500 From: Jeff Woods [EMAIL PROTECTED] Subject: Re: Mersenne: pi 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. 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.
