JSH: Twin primes
in [Math]
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From: marcus_b on 7 Feb 2010 15:01 On Feb 7, 12:46 pm, William Hughes <wpihug...(a)hotmail.com> wrote: > 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 Well, it should be nonzero. There is after all one prime triple. But your point, which has gone right over the Harris head in a mighty whoosh, is that Harris's logic applies to show that the probability of prime triples of any size is nonzero, i.e., there are infinitely many of them. He doesn't see this. He doesn't see it. The logic somehow escapes him. Not because he can't get it, but because he doesn't want to. Marcus.
From: master1729 on 7 Feb 2010 07:37 > On Feb 5, 5:20 pm, spudnik <Space...(a)hotmail.com> > wrote: > > 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). > > They ARE independent. There is just clipping > behavior for the bigger > primes for their higher residues which is so easy to > see. > > Between 5^2 and 7^2, there are 6 primes. The > probability then is given > by: > > prob = ((5-2)/(5-1))*((3-2)/(3-1) = (3/4)*(1/2) = > 0.375 > > And 6*0.375 = 2.25 so you expect 2 twin primes in > that interval. > > The primes are 29, 31, 37, 41, 43, 47 and you'll > notice, two twin > primes as predicted: 29,31 and 41, 43. > > However, there is an issue which shifts the > probability slightly. > > If you go into the actual residues it jumps out at > you: > > 29, 31, 37, 41, 43, 47 > > mod 3: 2, 1, 1, 2, 1, 2 > mod 5: 4, 1, 2, 1, 3, 2 > > Here all the residues for 5 were in evidence so the > count came out > right, but for random it should have been possible > for ALL the > residues mod 5 to be 4, but it's not because with 6 > primes there isn't > enough space in the interval--4*5 = 20, but 48-25=24, > where only 12 > are odd and only 6 are primes. So the probability is > actually off! A > scenario where all residues are 4 is precluded by the > size of the > interval for the larger prime. > > That will tend to over-count because the higher > residues are less > likely to occur because they cannot fit. > > They cannot fit. > > So some probabilities are dropped to zero because > they're impossible > in the interval. > > That means that 2 as a lower residue is more likely > to occur which > means the twin prime count will be lower. > > Trivial to a real researcher. > > So the over count is easily explained and > probabilistic behavior is > still the only answer that makes sense. > > Of course if you're some loser who can't do real > research to save your > life it's convenient to claim otherwise for > government grants, but > then you're just on white collar welfare. > > White collar welfare. > > But if you ARE some pretender trying to be a > mathematician what else > can you do? > > If you tell the truth your only option is to quit > pretending and leave > the field as you can't do real research, so primes > are a draw for the > fakes. > > > James Harris > there is indeed 'overcount' but mertens already considered it.
From: master1729 on 7 Feb 2010 07:40 > > But if you say 2+2 = 5, it doesn't matter how many > people agree with > you. that actually sounds like ullrich :) > > You're still all wrong. back to harris sounds. > > > James Harris >
From: master1729 on 7 Feb 2010 07:46 > On Thu, 4 Feb 2010 22:10:08 -0800 (PST), Arturo > Magidin > <magidin(a)member.ams.org> wrote: > > >On Feb 4, 7:25 pm, Rick Decker > <rdec...(a)hamilton.edu> wrote: > >> Arturo Magidin wrote: > > > >> > 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%. > >> > >> While the equidistribution of primes in residue > classes mod p is > >> indeed an established fact (so no need to make it > an axiom), it's an > >> asymptotic result and will frequently lead to > errors when applied > >> to individual cases, as the ones that lead to the > what I've generously > >> called the Harris function. > > > >Yes, but there is more than that. Sarnak and > Rubinstein proved in 1995 > >that if you look at the sum of (1/x) over all the x > up to N for which > >there are more primes of the form 4n+1 than of the > form 4n+3, and then > >take the limit of as N--->oo of this sum divided by > ln(N), then the > >limit exists and is equal to 0.9959.... In a sense, > about 99.59% of > >the time there are more primes of the form 4n+1 up > to x than of the > >form 4n+3. (This has to do with the fact that 1 is a > square mod 4 and > >3 is not, by the way). The set of such X's does not > have asymptotic > >density (the limit does not exist); and > Hardy-Littlewood tells you > >that the race will change "leaders" infinitely > often. But the > >deviations "favor" one of the congruence classes > over the other in a > >rather striking way. > > 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? > > > > in most cases , because JSH wasnt born yet ? :)
From: master1729 on 7 Feb 2010 09:00
> could some one, please, > describe what the axiom says, > in non-HSJ-speak? who is HSJ ? :) > > --les OEuvres! > http://wlym.com > > thus quoth: > How recent changes in the solar dynamo are affecting > our > interplanetary 'weather' > by J.G. Luhmann global warming is a fact. btw its very cold outside ...hmmm .... > > The photospheric magnetic field provides the key > boundary conditions > for the interplanetary medium, including the solar > wind plasma and the > interplanetary magnetic field. Thus any changes in > the solar interior > that affect the emergence, dispersion, and decay of > active region > magnetic fields, or the evolution of the those > fields, can have > locally measurable effects. Each solar cycle for > which we have > sufficient solar and in-situ observations has been > somewhat different, > but in the most recent cycle the solar dynamo has > produced a generally > weaker photospheric field together with a spatial > distribution of > photospheric flux that differs from the previous two. > The results give > us a better appreciation for how the dynamo affects > our 'space > weather', together with any other effects that might > be produced by > related solar irradiance changes. c + v = c or whatever ;) > > Tuesday, 02 February 2010 > 3853 Slichter Hall > Refreshments at 3:45 PM refreshment at 0:59. |