JSH: Twin primes
in [Math]
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From: MichaelW on 5 Feb 2010 07:23 On Fri, 05 Feb 2010 02:44:43 -0800, Rotwang wrote: > On 5 Feb, 05:07, MichaelW <ms...(a)tpg.com.au> wrote: >> >> [a whole load of stuff, then this] >> >> As you can see the ratio of the prediction to the actual value stays >> pretty close to about 1.12. > > This number is rather familiar to those of use who were following > James's threads a few years back. See the following article[1], and if > you're after more info then it's worth looking at Tim Peters' posts from > around August 2006, especially those that mention "Mertens". > > [1] http://groups.google.co.uk/group/alt.math/msg/95688f21176359b1 Thanks Rotwang, that was an interesting read. I am still wondering if there is any relationship between the number 1.123... and the twin primes constant? I am not surprised that James is claiming victory with an idea that a simple program can and has disposed of long before. This is a known behaviour. I found an excellent quote from that era by Tim Peters which I will reproduce here since it really expanded my understanding of what is going on: <TimP> The fatal flaws in your "naive" probability models are explained in Hans Riesel's book "Prime Numbers and Computer Methods for Factorization", in chapter "Subtleties in the Distribution of Primes". You're not the first person to get sucked into building on a fatally naive model, and won't be the last. The primes don't actually act the way you assume they act (to get the probability of N events occurring simultaneously you simply multiply the probabilities of the N individual events, but that's valid only if the events are /independent/, and while I grant that it's not / intuitively/ obvious, the events here are not independent; Mertens's Theorem is in fact a rigorous proof that they're not independent in the sense you need them to be). </TimP> It is sad that posts such as yours and Tim's (informative, providing references, clear, the product of hard work and a passion for rigorous thought) are so rare these days; certainly beyond my lowly skills. Thanks, Michael W.
From: William Hughes on 5 Feb 2010 09:32 On Feb 4, 11:55 pm, JSH <jst...(a)gmail.com> wrote: > On Feb 4, 5:25 pm, Rick Decker <rdec...(a)hamilton.edu> wrote: > > > > > Arturo Magidin wrote: > > > On Feb 4, 2:01 pm, marcus_b <marcus_bruck...(a)yahoo.com> wrote: > > > >> Notice that no one here is saying you are wrong. Your > > >> statement is imprecise - it is really about asymptotic > > >> frequencies of residues - but it looks to me like it has > > >> a good chance of being true, and I don't think anyone has > > >> proved it, even for a small prime like 3. I think it would be a > > >> valuable theorem if it is true. > > > > There is the quantitative form of Dirichlet's Theorem, and > > > Chebotarev's Density Theorem. If N>=2 and gcd(a,N)=1, then the density > > > of primes congruent to a modulo N is asymptotic to phi(N), where phi > > > is Euler's totient function. This follows from Chebotarev (as someone > > > pointed out ot me recently) by looking at the cyclotomic extension > > > modulo N. > > > > For N=3, it says that asymptotically, "half" the primes are congruent > > > to 1 mod 3, and half are congruent to 2 mod 3. For N=4, "half" are > > > congruent to 1 mod 4, half are congruent to 3 mod 4. (But on that > > > note, as I mentioned recently as well, there are certain measures by > > > which it makes sense that there are "usually" far more primes > > > congruent to one of the two classes, can't remember which, in the > > > sense that if you consider the x for which the race up to x is lead by > > > those congruent to 1 mod 4, then a certain density measure for those x > > > gives you a deviation from 0.5). > > > And it is just this race behavior that messes up what we might call > > the Harris function: > > > H(q) = product(1 - 1/(p-1), p prime, 2 < p < \sqrt(q+2)) > > > which he erroneously claims is the probability that q + 2 will be a > > prime, given that q is. > > > He's done some examples of this, computing the predicted and actual > > number of twin primes in the interval (p_i)^2 ... (p_{i+1})^2 but as > > often happens, James falls into the fallacy of the Law of Small Numbers.. > > For example, when we take the two consecutive primes 5471 and 5523 > > we find that there are 33282 primes in the interval 5471^2 .. 5523^2 > > and of these 2605 are twin primes. However, James' result would predict > > that H(p) = 0.043.. where p is the largest prime smaller than 54721 > > so he would predict that there would be 0.043 * 33282 = 1433 (approx) > > twin primes, a relative error of almost 82%. > > So? What's the probability of that event? Very difficult to calculate unless you incorrectly assume independence. It is interesting to note that your overestimate of the number of primes is about equal to your overestimate of the number of double primes. This would suggest that you can figure out the number of double primes by taking the known prime probability and making an independence assumption. Hughesless conjecture: You can figure out the number of triple primes by taking the known prime probability and making an independence assumption. - William Hughes
From: marcus_b on 5 Feb 2010 17:47 On Feb 4, 9:55 pm, JSH <jst...(a)gmail.com> wrote: > On Feb 4, 5:25 pm, Rick Decker <rdec...(a)hamilton.edu> wrote: > > > > > Arturo Magidin wrote: > > > On Feb 4, 2:01 pm, marcus_b <marcus_bruck...(a)yahoo.com> wrote: > > > >> Notice that no one here is saying you are wrong. Your > > >> statement is imprecise - it is really about asymptotic > > >> frequencies of residues - but it looks to me like it has > > >> a good chance of being true, and I don't think anyone has > > >> proved it, even for a small prime like 3. I think it would be a > > >> valuable theorem if it is true. > > > > There is the quantitative form of Dirichlet's Theorem, and > > > Chebotarev's Density Theorem. If N>=2 and gcd(a,N)=1, then the density > > > of primes congruent to a modulo N is asymptotic to phi(N), where phi > > > is Euler's totient function. This follows from Chebotarev (as someone > > > pointed out ot me recently) by looking at the cyclotomic extension > > > modulo N. > > > > For N=3, it says that asymptotically, "half" the primes are congruent > > > to 1 mod 3, and half are congruent to 2 mod 3. For N=4, "half" are > > > congruent to 1 mod 4, half are congruent to 3 mod 4. (But on that > > > note, as I mentioned recently as well, there are certain measures by > > > which it makes sense that there are "usually" far more primes > > > congruent to one of the two classes, can't remember which, in the > > > sense that if you consider the x for which the race up to x is lead by > > > those congruent to 1 mod 4, then a certain density measure for those x > > > gives you a deviation from 0.5). > > > And it is just this race behavior that messes up what we might call > > the Harris function: > > > H(q) = product(1 - 1/(p-1), p prime, 2 < p < \sqrt(q+2)) > > > which he erroneously claims is the probability that q + 2 will be a > > prime, given that q is. > > > He's done some examples of this, computing the predicted and actual > > number of twin primes in the interval (p_i)^2 ... (p_{i+1})^2 but as > > often happens, James falls into the fallacy of the Law of Small Numbers. > > For example, when we take the two consecutive primes 5471 and 5523 > > we find that there are 33282 primes in the interval 5471^2 .. 5523^2 > > and of these 2605 are twin primes. However, James' result would predict > > that H(p) = 0.043.. where p is the largest prime smaller than 54721 > > so he would predict that there would be 0.043 * 33282 = 1433 (approx) > > twin primes, a relative error of almost 82%. > > So? What's the probability of that event? > Probability of seeing 2605, when the expected number is 1433? EXTREMELY small. My program estimates it to be 3.90 x 10^(-179). > The expectation for 10 coin flips is 5 heads and 5 evens. > > Giving a case of 10 heads does not change that. > > Your post intrigues me. It's fascinating, right? > It almost spits in the face of the readership > of sci.math as if none of them know probability. > What makes you think that? > It's like a slap in the face of the entire newsgroup. > It's more like a slap in the face to you. > An expression of contempt for their basic intelligence. > Yeah. But for yours. Marcus. > James Harris
From: spudnik on 5 Feb 2010 20:20 well, that seems rather to dyspoze of the whole issue, viz-a-vu. and, like I said, in January, that Magadin said, about primes of the form 4n +/- 1. thus quoth: Mertens's Theorem is in fact a rigorous proof that they're not independent in the sense you need them to be). --les OEuvres! http://wlym.com --Stop the quagmire retread in Afghanistan; see "The Most Dangerous Man" on Feb.12 opening at Music Hall Theater on Wilshire in Beverley Hills! http://www.laemmle.com/viewmovie.php?mid=5177
From: spudnik on 5 Feb 2010 20:25
yeah; math is so *over*. > You guys talking about all these detailed results that you claim > people have actually proved must be lying. How could anyone > possbly have done work in this area before JSH discovered the > whole thing?- Hide quoted text - thus: well, that seems rather to dyspoze of the whole issue, viz-a-vu. and, like I said, in January, like that Magadin said, about primes of the form 4n +/- 1 (-1 is not a second power mod 4 ?-) thus quoth: Mertens's Theorem is in fact a rigorous proof that they're not independent in the sense you need them to be). --les OEuvres! http://wlym.com --Stop the quagmire retread in Afghanistan; see "The Most Dangerous Man" on Feb.12 opening at Music Hall Theater on Wilshire in Beverley Hills! http://www.laemmle.com/viewmovie.php?mid=5177 |