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Message: 5 >Date: Sun, 01 Oct 2006 09:22:10 +0200 >From: [EMAIL PROTECTED] >Subject: [Prime] Possible algorithm for predicting the sequence number > for each prime? >To: [email protected] >Cc: [EMAIL PROTECTED] >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain; charset=ISO-8859-1 > >Hi, > >Robert Betts said: "I meant to add that all the prime numbers are known to be >distributed randomly--I repeat--randomly along the real line." > >Why "randomly" ? >If I remember well, a set of numbers is random if the shortest way to describe >it is to provide the list of these numbers (there is no algorithm to compute >them, and knowing the first N numbers does not help to predict number numbered >N+1). >Since the Eratosthem sieve algorithm can produce the list of all prime numbers, >prime numbers do not appear randomly. >Since knowing all primes below sqrt(P) can be used to prove that P is prime or >not, prime numbers do not appear randomly. >Large prime numbers seems to appear randomly to us because they would require >computers and time as big as our Universe. > >So: prime numbers are not distributed randomly. > >See: >http://en.wikipedia.org/wiki/Random_number >http://en.wikipedia.org/wiki/Chaitin%E2%80%93Kolmogorov_randomness >http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer.html > >Regards, > >Tony > > >------------------------------ > >Message: 6 >Date: Sun, 1 Oct 2006 17:40:55 +0000 >From: Brian Beesley <[EMAIL PROTECTED]> >Subject: Re: [Prime] Possible algorithm for predicting the sequence > number for each prime? >To: The Great Internet Mersenne Prime Search list <[email protected]> >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain; charset="iso-8859-1" > >On Sunday 01 October 2006 07:22, [EMAIL PROTECTED] wrote: > >> Robert Betts said: "I meant to add that all the prime numbers are known to >> be distributed randomly--I repeat--randomly along the real line." > >That's obviously wrong - though (over reasonable intervals) the density of >prime numbers is approximately uniformly distributed when plotted against the >_logarithm_ of the interval midpoint. >> >> Why "randomly" ? >> If I remember well, a set of numbers is random if the shortest way to >> describe it is to provide the list of these numbers (there is no algorithm >> to compute them, and knowing the first N numbers does not help to predict >> number numbered N+1). ............................................................................................................................................................ [Response] By the term below "sequence number," I mean 1 is the sequence number for the prime 2, and 2 for 3, and 3 for 5, 4 for 7, 5 for 11, 6 for 13, and so on, where the sequence is 2, 3, 5, 7, 11, 13, 17, .... If Poapst's algorithm uses the Prime Number Theorem function Pi(x) for large input number prime p, then a subroutine or other algorithm could assign a number for p to give its position in the prime number sequence 2, 3, 5, ..., p, provided each and every prime less than p is known. Therefore I agree that if A. Poapst's algorithm does use the Prime Number Theorem, he could add an additional algorithm that could assign a probable sequence number for p, but GIVEN that ALL the primes < p also are known. But what if they are NOT known? How can Poapst's algorithm be used to assign sequence numbers for primes that are not yet known to exist? However, even this--using the Prime Number Theorem--does not show that the primes are not randomly distributed along the real line. If prime numbers were not randomly distributed along the real line, and if I am not mistaken on this, a lot of unsolved conjectures would be answered. Any prime easily could be found if they are not randomly distributed, including the twin primes. Let me illustrate. Suppose someone was to say, "What is the 1000-th positive even integer? That's easy. Just use an algorithm Each Even Number = 2n, n a positive integer, and plug in 1000 as n in the iteration, you get 2000 as the 1000-th positive even integer. But can you do this with all prime numbers? Let me raise the issue on this List as a formal question: QUESTION: Does there exist an iteration algorithm F for any input prime p, such that, for any such prime number p appearing in the sequence, 2, 3, 5, 7, 11, 13, 17, ..., p, ..., we have p ------------> F(p) = i, where "i" is the position of the prime number p in the sequence 2, 3, 5, ....? What about without using the Prime Number Theorem? If the primes are not randomly distributed on the real line, then it would seem you would not need the Prime Number Theorem function Pi() to find out what number of prime the prime p is in the infinite sequence of primes 2, 3, 5, ..., p, ..... Let me illustrate what I meant when I said primes were randomly distributed along the real line. Note when I said "randomly distributed" I don't mean they have a "uniform distribution". That is not what I mean when I say they are "randomly distributed." The exact nature of the distribution of prime numbers on the real line is a big area of research. Some mathematicians and quantum theorists think the random distribution of primes is related to quantum mechanics, quantum chaos and quantum computation. Suppose, P_{1} was the largest prime known to exist, discovered perhaps by a computer search using the sieve of Eratosthenes (a program using this would run very slow, I think!), or else maybe Pollard (p-1)--factorization or the Lucas-Lehmer test. Let P_{2} > P_{1} be an even much larger prime number, but such that P_{1} and P_{2} are NOT consecutive prime numbers. Suppose someone discovered the prime P_{2}, perhaps as I said with the Lucas-Lehmer test or using some other algorithm, and that this prime never was known to exist before. Suppose further that there still are unknown prime numbers existing between P_{1} and P_{2} > P_{1}? Question: Could Allen Poapst's algorithm find the sequence number for a recently discovered prime P_{2}, given that all the prime numbers between P_{1} and P_{2} on the real line still are not known? Of course, like some have posted on this thread, for any KNOWN, large enough positive integer or large enough positive real number n, the number of primes < n, or Pi(n), could be found by the Prime Number Theorem, given good software commands that use the Prime Number Theorem like the function primes() in MATLAB. But these algorithms can only be used to give the number of primes less than the integer n, which is known. To my knowledge, no algorithm yet outputs the sequence number for any input P_{2} in its position in the infinite sequence of prime numbers 2, 3, 5, 7, 11, 13, ..., P_{1}, ..., P_{2}, ..... The function primes() in MATLAB does give you the column each of the primes < n is in, and from that you can deduce what its sequence number "i" is. But I am not familiar with any software that can output "p is the i-th prime number," if p were NEVER known to exist before and if there are still unknown primes smaller than p. For example all the prime numbers for the MATLAB primes() function are stored in a matrix. But what about those larger, unknown primes p that MATLAB doesn't know exists? What about the primes smaller than p but still large, that MATLAB, MATHEMATICA or some other software package doesn't know exists? Suppose Poapst uses MATLAB or MATHEMATICA to input a huge, unknown prime p, but these software programs cannot output information about p, because there are other large, unknown primes < p that still are unknown to everyone? Please note: Pi(P_{2}) would, in fact, tell you the number of primes less than P_{2}. But Pi(P_{2}) does NOT tell you what number in the entire infinite sequence of prime numbers 2, 3, 5, ... P_{2] has. Also Pi(x) is fairly accurate but only up to big-oh precision. What would be the sequence numbers for all the unknown primes still lying between P_{1} and P_{2}? Pi(P_{2}) only gives you the number of primes less than P_{2}, not the position of each unknown prime in the finite subsequence of all primes less than P_{2}, including those lying between P_{1} and P_{2] that are not known. This precisely is what I mean. Granted, if P_{2} were large enough, you could use the value of Pi(P_{2}) to label yourself with sequence numbers, all the possible unknown primes between P_{1} and P_{2}, but computing with Pi(P_{2}) itself is NOT an algorithm that gives you these exact numbers in the aequence, I mean P_{1} is prime number i, P_{2] is prime number k, ..., What about all the primes lying between P_{1] and P_{2} that no one knows about? How can you know the number in the sequence for P_{2} when you don't know what the primes are between P_{1] and P_{2]? This is what I meant. But if Allen Poapst's algorithm does combine the Prime Number Theorem function Pi(x) along with some other subroutine or algorithm to assign the right number to P_{2} in the sequence of primes, then perhaps he does have something novel, IF it is used for primes p for which all primes smaller than p also ARE known, such as is the case when you use MATHEMATICA or MATLAB. So, given some extremely large prime number, hitherto unknown, say one with 10^{9} digits in it, and such that there are other primes smaller than it that still are unknown, could Allen Poapst's algorithm be used to determine what number it has in the sequence of prime numbers? I say that is unlikely, but if someone knows an algorithm by which it could be done, I would appreciate knowing about it. Right now GIMPS is looking for a 10 million digit Mersenne prime. Suppose this unknown, 10 million digit Mersenne prime is 2^{q} - 1. What if there are OTHER primes SMALLER than 2^{q} - 1, that are so huge no one has found them yet? By assumption, if Poapst's algorithm does what he and other listmembers believe, shouldn't it be possible to tell what number prime 2^{q} - 1 is in the sequence of prime numbers inclusive from 2 to 2^{q} - 1, even though those primes smaller than 2^{q} - 1 still are unknown? Shouldn't also it be possible for Poapst's algorithm to tell what the numbers for each unknown prime < 2^{q} - 1 is? I suspect this is not possible, but I still am open to contrary arguments. By the way, as I mentioned, there seems to be some connection between the distribution of primes on the real line, and quantum computation. Robert Betts Graduate Student, Mathematics (Alumnus, 2002): University of Massachusetts Department of Mathematics and Science Science Building 100 Morrissey Blvd. Boston, MA USA 02125 -----Original Message----- >From: [EMAIL PROTECTED] >Sent: Oct 1, 2006 2:51 PM >To: [email protected] >Subject: Prime Digest, Vol 30, Issue 1 > >Send Prime mailing list submissions to > [email protected] > >To subscribe or unsubscribe via the World Wide Web, visit > http://hogranch.com/mailman/listinfo/prime >or, via email, send a message with subject or body 'help' to > [EMAIL PROTECTED] > >You can reach the person managing the list at > [EMAIL PROTECTED] > >When replying, please edit your Subject line so it is more specific >than "Re: Contents of Prime digest..." > > >Today's Topics: > > 1. Possible algorithm for predicting the sequence number for > each prime? ([EMAIL PROTECTED]) > 2. Fwd: Prime sequence formula (Ian Halliday) > 3. Possible algorithm for predicting the sequence number for > each prime? ([EMAIL PROTECTED]) > 4. Prime sequence formula (Nacho) > 5. Possible algorithm for predicting the sequence number for > each prime? ([EMAIL PROTECTED]) > 6. Re: Possible algorithm for predicting the sequence number for > each prime? (Brian Beesley) > 7. Re: Prime sequence formula (Brian Beesley) > 8. Re: Possible algorithm for predicting the sequence number for > each prime? (Paul Leyland) > 9. Re: Prime sequence formula (Soo Reams) > 10. Re: Possible algorithm for predicting the sequence number for > each prime? (Andy Roland) > > >---------------------------------------------------------------------- > >Message: 1 >Date: Sat, 30 Sep 2006 18:25:02 -0400 (GMT-04:00) >From: "[EMAIL PROTECTED]" <[EMAIL PROTECTED]> >Subject: [Prime] Possible algorithm for predicting the sequence number > for each prime? >To: [email protected] >Cc: [EMAIL PROTECTED] >Message-ID: > <[EMAIL PROTECTED]> > >Content-Type: text/plain; charset=us-ascii > > >---------------------------------------------------------------------- >[Previous post by Allen Poapst follows] > > > >Hi, my name is Allen Poapst, attending Brock University, St. Catharines ON, >Canada. I was searching up prime sequence formula on the internet, and >happened to find your name and e-mail. > >Some sort of GIMPS 10,000$ thing, I have no clue what this is, is it a >reward for finding the formula? > >If so, I have found a formula for the prime sequence which I am trying to >get published through the university I am attending. Yes I know this is a >hotmail account and an informal letter, but it is just for inquiry purposes. > >Though the formula I have works in the following manner: > >f(x) = x.... > >x f(x) >2 1 >3 2 >5 3 >7 4 >11 5 >13 6 >. . >. . >. . >200,000,093 -> 11,078,945th prime (using my program that simulates my >formula, this calculation on my notebook takes 11 seconds) > >etc... until infinite, although I have only tested my formula to >approximately 5 million, using a computer simulation, the logic of the >formula makes sense for all numbers, since I found a pattern that my formula >exploits. So if you give a prime number into my formula, it will pop out >where it is in the prime sequence, though I have not found a formula that >works the other way around, where you give the position in the sequence and >the formula pops out the prime number. > >A professor that is helping me publish my formula at the university I am >attending says that he knows of no formula that is known for what I have >found. > >Can you confirm this? And also would my formula acctually be worth 10,000$, >that would be great if it was actually worth money. > >A response ASAP would be appreciated, and thank you for your time, I am >really excited about this formula if it is unknown and I discovered it. > >Thank you, > >Allen Poapst, Brock University > >My e-mails: >[EMAIL PROTECTED] >[EMAIL PROTECTED] >[EMAIL PROTECTED] > >..................................................................................................................................................................... > > > >[Response by [email protected] listmember Robert Betts] > > > >The actual distribution of the prime numbers along the real line is still >unknown, that is, as far as I know. In fact if the actual distribution was >known, I believe perhaps it indeed might be possible to write a computer >algorithm or a "formula" such as you state, but to predict the probability any >given prime had a certain sequence number such as you say your formula does, >given that these sequence numbers are assigned for each prime, for example >sequence number 1 for 2, sequence number 2 for 3, sequence number 3 for 5..... > >Let x be a positive real number. What is known for sure about where a certain >number of primes are, is known from the Prime Number Theorem, which says that >the number of primes less than x, denoted by pi(x), is > > >pi(x) = x/ln(x). > > >So if I am not mistaken, (and perhaps a listmember who is a bona fide >mathematics professor can help me here because I am just a graduate student), >for your formula really to work would require knowledge on the actual >distribution of the prime numbers along the real line. Also it would depend on >someone actually proving the Riemann Hypothesis, that the Riemann zeta >function has all its nontrivial zeroes along the line Re(z) = 1/2 in the >complex plane. > >If you'd like to read up on this (like I have), you might want to obtain: > > >1. Riemann's Zeta Function, H.M. Edwards, Dover Publications, NY, 2001. >(You'll find a proof of the Prime Number Theorem and an appendix with >Riemann's notes on this topic, in this book) > > >2. The Art of Computer Programming, Volume II, Donald Knuth >(excellent book! unfortunately I cannot remember the publisher for this, but >the book is in my alma mater university library. You should be able to find it >at your school's library). > > >It's nice to hear your professor is helping you with publication. I'm envious. >I wish I was so lucky. > > >Robert Betts >Graduate Student, Mathematics >(Alumnus, 2002): University of Massachusetts >Department of Mathematics and Science >Science Building >100 Morrissey Blvd. >Boston, MA USA 02125 > > > > > > > > >------------------------------ > >Message: 2 >Date: Sat, 30 Sep 2006 23:35:29 +0100 >From: "Ian Halliday" <[EMAIL PROTECTED]> >Subject: [Prime] Fwd: Prime sequence formula >To: "The Great Internet Mersenne Prime Search list" > <[email protected]> >Message-ID: > <[EMAIL PROTECTED]> >Content-Type: text/plain; charset=ISO-8859-1; format=flowed > >I have engaged in the following email discussion with Allen. I hope >that most of you here grasp what I am doing... > >Ian > >Ian W Halliday, BA Hons, SA Fin, ATMG, CL >+44 772 546 2965 (GMT+1) >https://www.linkedin.com/in/ianwhalliday > >---------- Forwarded message ---------- >From: Allen Poapst <[EMAIL PROTECTED]> >Date: 30-Sep-2006 20:38 >Subject: Re: [Prime] Prime sequence formula >To: Ian Halliday <[EMAIL PROTECTED]> > >On 9/28/06, Ian Halliday <[EMAIL PROTECTED]> wrote: >> Hi Allen, >> >> You'll probably know by now that your email was forwarded to a list of >> people interested in prime numbers. >> Obviously I don't want to know the formula you have, because you'll >> want to keep it secret for the time being, but I wonder whether you >> can give me some indication of what sort of formula it is. >> Is it a polynomial like a + bx + cx^2 + dx^3 ... >> or an iteration, where f(n+1) is dependent on f(n) >> or something with exponentials, or sines or cosines >> or something more exotic? >> I'd be interested in looking into your formula in some detail, as it >> would be a very significant piece of mathematics, but probably there >> will be lots of people who will dismiss it because this sort of thing >> has never been done before. It's the sort of thing I was interested in >> when I was studying number theory in the early 1980s. Of course, >> computer power wasn't so easily available for checking my work back >> then... >> I hope to hear from you shortly. >> >> Regards, >> >> Ian > >Hi Ian, > >Thank you for your interest. The professor I am with right now to >publish this says that it would be a great addition to mathematics, >and wants me to publish it ASAP, but I am currently getting the >forumla as elegant as possible and explained in detail first, since it >is pretty easy to understand the concept, yet hard to write down as a >formula. I would like to keep this formula secret, though it is not >polynomial, no sins or cosins, it does have some recursive elements in >the formula. > >I came up with this formula for an assignment in one of my classes >last winter, when I needed to look into prime numbers, it only took me >a week to come up with it. Then I tested it with my program and it >works up to at least 5 million. I'll e-mail it in detail once it is >published, because it would need to be proved by someone. > > >------------------------------ > >Message: 3 >Date: Sat, 30 Sep 2006 18:49:27 -0400 (GMT-04:00) >From: "[EMAIL PROTECTED]" <[EMAIL PROTECTED]> >Subject: [Prime] Possible algorithm for predicting the sequence number > for each prime? >To: [email protected] >Cc: [EMAIL PROTECTED] >Message-ID: > <[EMAIL PROTECTED]> > >Content-Type: text/plain; charset=us-ascii > >[Addendum] > > > >I meant to add that all the prime numbers are known to be distributed >randomly--I repeat--randomly along the real line. So it is highly improbable >that some algorithm or formula at present with the prime number distribution >not understood (if I am not mistaken about this), could accurately predict >what number any given prime might have within the prime number sequence. But I >am interested on what some professors in the field might say about this topic. >Can such an "indexing" formula ever exist? I don't know. > >Given that Gimps's next Mersenne prime were found and given to you without you >knowing its value in advance, could your formula actually predict its sequence >number, given that the prime distribution still is not sufficiently >understood? This is what I meant by "highly improbable". Your claim that such >a formula exists seems to me to be more of an example for a Zero knowledge >proof. > >Robert Betts > > >------------------------------ > >Message: 4 >Date: Sun, 01 Oct 2006 09:16:44 +0200 >From: Nacho <[EMAIL PROTECTED]> >Subject: [Prime] Prime sequence formula >To: [email protected], [EMAIL PROTECTED] >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain; charset=us-ascii; format=flowed > >Hello everyone. > >I'm a little surprised that nobody said it, but such function is known. >It is called Pi(x), and gives the number of primes less than x. The main >problem is the implementation of Pi(x), of course. > >Maybe the algorithm developed by Allen is new, but there are some other >algorithms used. > >For example, the software Mathematica implements the function >PrimePi[n], and the help says: > >"PrimePi use sparse caching and sieving. For large a, the >Lagarias-Miller-Odlyzko algorithm for PrimePi is used, based on >asymptotic estimates of the density of primes." > >It works with an n up to about 8*10^13 and it is very fast. > >For example, with the number of Allen, (200,000,093): > >PrimePi[200000093] //Timing >{0. Second, 11078945} > >You can find much more information and references here: > >http://mathworld.wolfram.com/PrimeCountingFunction.html > >Regards. > > > >------------------------------ > >Message: 5 >Date: Sun, 01 Oct 2006 09:22:10 +0200 >From: [EMAIL PROTECTED] >Subject: [Prime] Possible algorithm for predicting the sequence number > for each prime? >To: [email protected] >Cc: [EMAIL PROTECTED] >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain; charset=ISO-8859-1 > >Hi, > >Robert Betts said: "I meant to add that all the prime numbers are known to be >distributed randomly--I repeat--randomly along the real line." > >Why "randomly" ? >If I remember well, a set of numbers is random if the shortest way to describe >it is to provide the list of these numbers (there is no algorithm to compute >them, and knowing the first N numbers does not help to predict number numbered >N+1). >Since the Eratosthem sieve algorithm can produce the list of all prime numbers, >prime numbers do not appear randomly. >Since knowing all primes below sqrt(P) can be used to prove that P is prime or >not, prime numbers do not appear randomly. >Large prime numbers seems to appear randomly to us because they would require >computers and time as big as our Universe. > >So: prime numbers are not distributed randomly. > >See: >http://en.wikipedia.org/wiki/Random_number >http://en.wikipedia.org/wiki/Chaitin%E2%80%93Kolmogorov_randomness >http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer.html > >Regards, > >Tony > > >------------------------------ > >Message: 6 >Date: Sun, 1 Oct 2006 17:40:55 +0000 >From: Brian Beesley <[EMAIL PROTECTED]> >Subject: Re: [Prime] Possible algorithm for predicting the sequence > number for each prime? >To: The Great Internet Mersenne Prime Search list <[email protected]> >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain; charset="iso-8859-1" > >On Sunday 01 October 2006 07:22, [EMAIL PROTECTED] wrote: > >> Robert Betts said: "I meant to add that all the prime numbers are known to >> be distributed randomly--I repeat--randomly along the real line." > >That's obviously wrong - though (over reasonable intervals) the density of >prime numbers is approximately uniformly distributed when plotted against the >_logarithm_ of the interval midpoint. >> >> Why "randomly" ? >> If I remember well, a set of numbers is random if the shortest way to >> describe it is to provide the list of these numbers (there is no algorithm >> to compute them, and knowing the first N numbers does not help to predict >> number numbered N+1). > >Ah, but knowledge of the first few prime numbers _does_ influence prediction >of later ones in the sequence ... if P is a prime number, we know that kP is >composite for all integers k > 1 (by definition!) > >> Since the Eratosthem sieve algorithm can produce the list of all prime >> numbers, prime numbers do not appear randomly. > >Yes. > >> Since knowing all primes below sqrt(P) can be used to prove that P is prime >> or not, prime numbers do not appear randomly. > >Yes. > >> Large prime numbers seems to appear randomly to us because they would >> require computers and time as big as our Universe. > >Sorry but I don't understand this. In fact the universe is finite in content >and time whereas the set of positive integers (and primes) is infinite so I >don't see how any computer could possibly be big or fast enough to tabulate >all primes. Not even a quantum computer. >> >> So: prime numbers are not distributed randomly. > >By definition: _NO_ particular list of numbers can _POSSIBLY_ be random! > >Regards >Brian Beesley > > >------------------------------ > >Message: 7 >Date: Sun, 1 Oct 2006 17:45:56 +0000 >From: Brian Beesley <[EMAIL PROTECTED]> >Subject: Re: [Prime] Prime sequence formula >To: The Great Internet Mersenne Prime Search list <[email protected]> >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain; charset="iso-8859-1" > >On Sunday 01 October 2006 07:16, Nacho wrote: >> Hello everyone. >> >> I'm a little surprised that nobody said it, but such function is known. >> It is called Pi(x), and gives the number of primes less than x. The main >> problem is the implementation of Pi(x), of course. > >Doesn't this depend on the (AFAIK unproved) Riemann hypothesis? >> >> Maybe the algorithm developed by Allen is new, but there are some other >> algorithms used. > >This is where the possible interest is. If Allen's algorithm turns out to be >genuinely different, proveably correct and returns the same results as Pi(x) >then there may be a handle on proving Riemann. Or even the extended Riemann >hypothesis! > >Regards >Brian Beesley > > >------------------------------ > >Message: 8 >Date: 01 Oct 2006 19:09:33 +0100 >From: Paul Leyland <[EMAIL PROTECTED]> >Subject: Re: [Prime] Possible algorithm for predicting the sequence > number for each prime? >To: The Great Internet Mersenne Prime Search list <[email protected]> >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain > >On Sun, 2006-10-01 at 18:40, Brian Beesley wrote: > >> Sorry but I don't understand this. In fact the universe is finite in content >> and time > >Are you sure of that? > >I look forward to reading your forthcoming paper where you prove that >claim. > > >Paul > > > > >------------------------------ > >Message: 9 >Date: Sun, 01 Oct 2006 19:28:05 +0100 >From: Soo Reams <[EMAIL PROTECTED]> >Subject: Re: [Prime] Prime sequence formula >To: The Great Internet Mersenne Prime Search list <[email protected]> >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain; charset=ISO-8859-1; format=flowed > >Brian Beesley wrote: >> On Sunday 01 October 2006 07:16, Nacho wrote: >> >>> Hello everyone. >>> >>> I'm a little surprised that nobody said it, but such function is known. >>> It is called Pi(x), and gives the number of primes less than x. The main >>> problem is the implementation of Pi(x), of course. >>> >> >> Doesn't this depend on the (AFAIK unproved) Riemann hypothesis? >> >Not certain of this, but I think Pi(x) is defined as the number of >primes less than or equal to x. The Riemann hypothesis is concerned with >a particular approximation to Pi(x), a more accurate one than Pi(x) = x >/ lg x. >>> Maybe the algorithm developed by Allen is new, but there are some other >>> algorithms used. >>> >> >> This is where the possible interest is. If Allen's algorithm turns out to be >> genuinely different, proveably correct and returns the same results as Pi(x) >> then there may be a handle on proving Riemann. Or even the extended Riemann >> hypothesis! >> >> Regards >> Brian Beesley >> >Hmm, unlikely. Allen's algorithm sounds like it's at least as >computationally intensive as finding the primes individually. > >Soo Reams > > > >------------------------------ > >Message: 10 >Date: Sun, 1 Oct 2006 20:51:11 +0200 >From: "Andy Roland" <[EMAIL PROTECTED]> >Subject: Re: [Prime] Possible algorithm for predicting the sequence > number for each prime? >To: <[EMAIL PROTECTED]>, "The Great Internet Mersenne Prime > Search list" <[email protected]> >Message-ID: <[EMAIL PROTECTED]> >Content-Type: text/plain; charset="iso-8859-1" > >Best appreciation of Paul's comments is in 'Internet speak' - lol > >Would add to that the impression one gets from Hawking and company currently >is that the universe possibly is finite in content but not in time - it could >go on for ever. That said the thinking on all this keeps changing >fundamentally every few years!! > ----- Original Message ----- > From: Paul Leyland > To: The Great Internet Mersenne Prime Search list > Sent: Sunday, October 01, 2006 8:09 PM > Subject: Re: [Prime] Possible algorithm for predicting the sequencenumber > for each prime? > > > On Sun, 2006-10-01 at 18:40, Brian Beesley wrote: > > > Sorry but I don't understand this. In fact the universe is finite in > content > > and time > > Are you sure of that? > > I look forward to reading your forthcoming paper where you prove that > claim. > > > Paul > > > _______________________________________________ > Prime mailing list > [email protected] > http://hogranch.com/mailman/listinfo/prime > > >------------------------------ > >_______________________________________________ >Prime mailing list >[email protected] >http://hogranch.com/mailman/listinfo/prime > > >End of Prime Digest, Vol 30, Issue 1 >************************************ _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
