JSH: Twin primes
in [Math]
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From: marcus_b on 4 Feb 2010 15:01 On Jan 31, 11:37 pm, JSH <jst...(a)gmail.com> wrote: > On Jan 31, 8:58 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > > > On 01/31/2010 11:29 PM, JSH wrote: > > > > On Jan 31, 8:11 pm, Joshua Cranmer<Pidgeo...(a)verizon.invalid> wrote: > > >> With the exception of the 13-17 range, your predicted number proves to > > >> be higher than the actual. I didn't have a larger list of twin primes to > > > > Meaningless. It's a probability result. > > > The deviations should be normally distributed about 0. I'm pointing out > > that they certainly don't *seem* to be, although 10 points is a bit > > scant to make certain accusations, especially as I don't care to do a > > proper statistical hypothesis test right now. > > The result follows from a proof which is based on what I now call the > prime residue axiom, which states that primes do not have a preference > for a residue modulo another prime. > This is actually interesting. It's not an axiom however. It's a conjecture. You cannot just add new axioms willy-nilly to the number system. If you do, you have the problem of proving consistency, i.e. whether when you add them, you get a contradiction. If you can derive a contradiction then the proposed 'axiom' must be false. 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. But it's not an axiom. Assuming it is true and then using it to "prove" other things is like what is often done with the Riemann Hypothesis. But the resulting "theorems" are not really theorems. Marcus. > Your position is like a person noticing ten coin flips that are heads > who wants to know if the coin is fair, as that seems really unlikely. > > But it can happen as anyone who flips a coin a lot knows. > > But that's human nature. Your brain craves patterns. > > > >> If you were serious about this work, why not code a program that checks > > >> the predicted and actual value of twin primes for the first few hundred > > >> of them? Extrapolating based on a few low values does not make a > > >> compelling argument. > > > > Wouldn't matter. There's too much research money in this area and the > > > twin primes conjecture is too famous. > > > That is a... naive view. I can't say that I've ever met a researcher so > > egotistical they would actively withhold research merely to keep > > research money. > > I've talked about this result before. > > I had it over 3 years ago. > > It doesn't matter what I discover. > > The reaction is always the same. > > > > The math community will not allow me to get research acceptance here > > > any more than with all my other research results. > > > Your other research results which seem to have been, in general, at best > > poorly written and explained and at worst outright incorrect. > > Like my twin primes probability result? > > > > Why don't I just go ahead and prove or disprove the Riemann Hypothesis > > > with my prime counting function? Why don't I just use my quadratic > > > residue results to try and make a factoring program? Why don't I just > > > implement my optimal path algorithm and see if it really does solve > > > the traveling salesman problem? > > > The algorithm whose incorrectness I demonstrated thrice? > > Well you better hope you did. Another person said he proved it wrong > as well. > > I haven't checked it. > > But if it does work then you can end up defined by nothing else but > your vocal rejection. > > If that algorithm does work then it has a value to the world that is > incalculable. > > Claiming it wrong and being wrong would be like trying to stop people > from ever learning calculus. > > Knowledge is funny that way. People like you think here is some kind > of personal battle. > > But when I'm right you're fighting humanity. > > That's not a nice thing to do. > > > > I defined mathematical proof. I discovered tautological spaces. > > > > I gave the best general method for simplifying binary quadratic > > > diophantine equations. > > > And I help develop widely-used software and have inadvertently > > publicized minor new features (I did not expect that post to be so > > widely distributed....). If I were more egotistical, I could also > > stretch my claims to say the development of a full static analysis > > toolkit (only a framework to build upon, and incomplete at that). > > And if you hadn't? > > If just my optimal path algorithm is correct then it can mean more > rapid technological advances, for the entire world. > > Ok, you say it's wrong. I hope you're right. > > As if you are wrong then you may have destroyed your future career and > I really do not know exactly why you have. > > > > Your opinion of me is as irrelevant as your ignorance of the > > > mathematics I understand. > > > ... Newton once said, "If I have seen a little further, it is by > > standing on the shoulders of giants." Could you be as humble as him? > > Newton shredded his foes. He was known during his time as a person > you did not cross. > > Later on he was in charge of the mint and had counterfeiters executed. > > History picks and chooses. It also re-writes as it sees fit. > > If you knew Newton the man, you would probably hate him. > > You certainly would not call him humble. > > > > You need bogus math to pay the bills. I don't. > > > I'm not a paid mathematician, and the last job I had that involved > > moderately complex mathematics was almost four years ago. I'm sure > > that's true of most people in this forum: most professors probably don't > > have the time to read newsgroups, and certainly not to respond. > > Which is why YOU for a moment acted as if the twin primes result was > interesting. > > I was talking about the people who do need it, to pay the bills. > > My optimal path algorithm is fun to me. I like it. I have no more > motivation to use it to force the issue with the traveling salesman > problem than I need to try and use the twin primes result, as I know > how fiercely people fight when their livelihoods are threatened. > > Quite a few math people do fake math. They lie about what they're > doing and the world lets them, for now. > > But when it stops, make no mistake, these people will fight to keep > the money flowing. > > And when they lose, they'll be angry. > > The truth is for the people who can afford it. > > James Harris
From: Arturo Magidin on 4 Feb 2010 15:13 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). -- Arturo Magidin
From: Rick Decker on 4 Feb 2010 20:25 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%. 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. Regards, Rick
From: JSH on 4 Feb 2010 22:55 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? 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 almost spits in the face of the readership of sci.math as if none of them know probability. It's like a slap in the face of the entire newsgroup. An expression of contempt for their basic intelligence. James Harris
From: Michael Press on 5 Feb 2010 00:06
In article <701ea136-d21c-4f22-95b3-5300d53d4f21(a)l24g2000prh.googlegroups.com>, amzoti <amzoti(a)gmail.com> wrote: > On Feb 2, 8:21 pm, JSH <jst...(a)gmail.com> wrote: > > On Feb 1, 5:32 pm, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > > > > On Feb 1, 9:16 pm, JSH <jst...(a)gmail.com> wrote: > > > > > > On Feb 1, 5:02 pm, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > > On Feb 1, 8:51 pm, JSH <jst...(a)gmail.com> wrote: > > > > > > > > The twin primes probability result is such an overwhelming one as > > > > > > mathematicians have been working for years building up data in support > > > > > > of it. > > > > > > > The GC result is such an overwhelming one as > > > > > mathematicians have been working for years building up data in support > > > > > of it. > > > > > > Except the prime residue axiom leads to a proof that Goldbach's > > > > Conjecture is false. > > > > > Wooosh! > > > > > The point is that both the GC and the "prime residue axiom" > > > are supported by lots of numerical evidence. > > > Why do you conclude that one is true and the other false? > > > > > - William Hughes > > > > It is self-evident that primes do not have a residue preference. > > > > For instance, why would 3 wish for primes to have 1 as a residue > > versus 2? I use "wish" deliberately to ask for intent. > > > > If there is no intent, then by what mechanism could such a preference > > result? > > > > That is what makes an axiom--self-evidence. > > > > In contrast Goldbach's Conjecture is not self-evident. It does not > > require intent to ask if every composite can be written as the product > > of 2 primes. > > > > But worse, with the prime residue axiom that is self-evident, you can > > disprove Goldbach's Conjecture. > > > > I notice that no one has asked how. > > > > Wow. Can you believe that? Not a single question as to how. > > > > Even as a mental exercise, one would think that some of you would be > > curious about how you disprove Goldbach's Conjecture with something as > > simple as saying that primes have no residue preference. > > > > But curiosity requires humanity. > > > > James Harris > > Can you tell me how to make a better martini using primes and > residues? > > Screw proving GC, I want better prime martinis! 1. Use prime liquor. 2. 1/3 vermouth, 2/3 gin. 3. Serve with a twist of lemon peel. -- Michael Press |