JSH: Twin primes
in [Math]
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From: William Hughes on 6 Feb 2010 19:15 On Feb 6, 7:47 pm, JSH <jst...(a)gmail.com> wrote: > On Feb 6, 2:43 pm, William Hughes <wpihug...(a)hotmail.com> wrote: > > > On Feb 6, 11:59 am, JSH <jst...(a)gmail.com> wrote: > > > > They ARE independent. > > > <snip> > > > >it should have been possible for ALL the > > > residues mod 5 to be 4, but it's not > > > So the residues are NOT independent. > > If there are only 6 primes in an interval from 25 to 49, then it's > just not possible for all the primes to be 4 mod 5, Wooosh! You have just said that the residues are not independent. <snip> > > Does it not bother you that using > > the exact same reasoning you use > > to show there are an infinite number > > of double primes we can show there > > are an infinite number of triple primes? > > The prime residue axiom states that > the primes do not care about their > residue modulo other primes, > which means you can calculate twin primes > probability easily enough The prime residue axiom states that the primes do not care about their residue modulo other primes, which means you can calculate triple primes probability easily enough. Does it not bother you that this calculation is wrong? - William Hughes
From: JSH on 7 Feb 2010 12:21 On Feb 5, 2:47 pm, marcus_b <marcus_bruck...(a)yahoo.com> wrote: > 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). Your program is wrong. Ask yourself, how many standard deviations is that from the mean, assuming a normal distribution? James Harris
From: JSH on 7 Feb 2010 12:23 On Feb 6, 4:15 pm, William Hughes <wpihug...(a)hotmail.com> wrote: > On Feb 6, 7:47 pm, JSH <jst...(a)gmail.com> wrote: > > > On Feb 6, 2:43 pm, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > On Feb 6, 11:59 am, JSH <jst...(a)gmail.com> wrote: > > > > > They ARE independent. > > > > <snip> > > > > >it should have been possible for ALL the > > > > residues mod 5 to be 4, but it's not > > > > So the residues are NOT independent. > > > If there are only 6 primes in an interval from 25 to 49, then it's > > just not possible for all the primes to be 4 mod 5, > > Wooosh! > > You have just said that the residues are not > independent. > > <snip> > > > > Does it not bother you that using > > > the exact same reasoning you use > > > to show there are an infinite number > > > of double primes we can show there > > > are an infinite number of triple primes? > > > The prime residue axiom states that > > the primes do not care about their > > residue modulo other primes, > > which means you can calculate twin primes > > probability easily enough > > The prime residue axiom states that > the primes do not care about their > residue modulo other primes, > which means you can calculate triple primes > probability easily enough. But the probability is shifted by excluded cases. 29, 31, 37, 41, 43, 47 mod 3: 2, 1, 1, 2, 1, 2 mod 5: 4, 1, 2, 1, 3, 2 Here there is no constraint for 3, but for 5, it's impossible for all the residues to be 4 within that interval. > Does it not bother you that this calculation is wrong? > > - William Hughes Wrong? Do you know what probability is? James Harris
From: marcus_b on 7 Feb 2010 13:34 On Feb 7, 11:21 am, JSH <jst...(a)gmail.com> wrote: > On Feb 5, 2:47 pm, marcus_b <marcus_bruck...(a)yahoo.com> wrote: > > > > > 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). > > Your program is wrong. Ask yourself, how many standard deviations is > that from the mean, assuming a normal distribution? > > James Harris You should assume a binomial distribution. But it doesn't matter a lot. 2605 is a little over 31 standard deviations away from 1433. Marcus.
From: William Hughes on 7 Feb 2010 13:46
On Feb 7, 1:23 pm, JSH <jst...(a)gmail.com> wrote: > On Feb 6, 4:15 pm, William Hughes <wpihug...(a)hotmail.com> wrote: <snip> > > Does it not bother you that this calculation is wrong? > > > - William Hughes > > Wrong? Yes wrong. You calculate a non-zero probability. - William Hughes |