JSH: Twin primes
in [Math]
|
From: JSH on 31 Jan 2010 21:35 On Jan 30, 10:53Â am, JSH <jst...(a)gmail.com> wrote: > One of the weirder things I discovered a while back was a resistance > to probabilistic explanations for some prime things where the easiest > area to see it boldly displayed is with twin primes probability. > > To understand fully, imagine that you accept that primes don't have a > preferred residue modulo themselves with other primes. Â For instance, > 3 has two potential residues modulo other primes: 1 and 2. Â Should it > prefer 1? Â Or maybe 2? Â No. Â Why would 3 care to lean towards either > residue? > > If so, then what residue a particular prime has mod 3 should be > random. > > Ok, so now let's get to twin primes. > > Here a trivial little result relating to twin primes as if x is prime > and greater than 3 the probability that x+2 is prime is given by: > > prob = ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*...*(1/2) > > where j is the number of primes up to sqrt(x+2), and p_j is the jth > prime, p_{j-1} is the prime before it and so forth. > > The result is easy as it is just multiplying the probability for each > prime that it is NOT true that > > x + 2 â¡ 0 mod p > > which probability is just the result of dividing one minus the number > of non-zero residues by the total number of residues together to get > the total probability that a prime plus 2 is also prime. > > So let's try it out. Â Between 5^2 and 7^2, there are 6 primes. Â 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. > > So that's a fun little thing where you can calculate easily when > you're bored or something and it works crazy well. Â Where it is all > just about a simple little idea that prime numbers aren't picking in > this simple way, and some of you of course know that what I've given > looks like a piece of Brun's constant. The nice thing about this result is that it is all about random and probability and you can SEE a piece of the equation I show in current literature on twin primes. They just don't explain it right. Now if you know that primes don't care about their residue modulo another prime, then there is no other reason available for the existence of twin primes. Gaps between primes are just about probability as the primes DO NOT CARE. So research in this area is stupid as there can be no other reason. That's what random means: no reason. I have used this example for years as I mulled the reality of the modern mathematical community beyond the Hollywood fantasy. Beautiful minds these are not! But it's also kind of sad as I know why they would rely on such crutches. Doing real mathematical research is HARD. So they cheat. I call it white collar welfare. James Harris
From: Joshua Cranmer on 31 Jan 2010 23:11 On 01/30/2010 01:53 PM, JSH wrote: > So let's try it out. 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. So let's try it out more: Range primes prob predict actual 3- 5 5 .5 -> 2.50 2 5- 7 6 .375 -> 2.25 2 7-11 15 .3125 -> 4.69 4 11-13 9 .28125 -> 2.53 2 13-17 22 .257813 -> 5.67 7 17-19 11 .241699 -> 2.66 2 19-23 27 .228271 -> 6.16 4 23-29 47 .217896 -> 10.24 8 29-31 16 .210114 -> 3.36 2 31-37 57 .203110 -> 11.58 11 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 count the actual numbers, so this table stops at 37, which is before what I think would be the interesting ranges 47-53 (the first pair of numbers where neither is a twin prime) and 59-61 (the first pair of twin primes after that). If you notice, 23 (the first non-twin prime) is involved in the two most "egregious" overestimates. 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. -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
From: JSH on 31 Jan 2010 23:29 On Jan 31, 8:11 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > On 01/30/2010 01:53 PM, JSH wrote: > > > So let's try it out. 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. > > So let's try it out more: > Range primes prob predict actual > 3- 5 5 .5 -> 2.50 2 > 5- 7 6 .375 -> 2.25 2 > 7-11 15 .3125 -> 4.69 4 > 11-13 9 .28125 -> 2.53 2 > 13-17 22 .257813 -> 5.67 7 > 17-19 11 .241699 -> 2.66 2 > 19-23 27 .228271 -> 6.16 4 > 23-29 47 .217896 -> 10.24 8 > 29-31 16 .210114 -> 3.36 2 > 31-37 57 .203110 -> 11.58 11 > > 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. Human nature is to try and find patterns in random, but it's just a brain habit. > count the actual numbers, so this table stops at 37, which is before > what I think would be the interesting ranges 47-53 (the first pair of > numbers where neither is a twin prime) and 59-61 (the first pair of twin > primes after that). If you notice, 23 (the first non-twin prime) is > involved in the two most "egregious" overestimates. By the prime residue axiom the primes DO NOT CARE about their residue modulo another prime so it's true randomness. There is no reason in it. Here's a bigger example: The probability that for a prime between 97^2 and 100^2 that adding 2 to it gives a prime is about 15.58% and there are 66 primes in that interval so there should be about 10 twin primes, and a quick count shows that there are: (9419, 9421), (9431, 9433), (9437, 9439), (9461, 9463), (9629, 9631), (9677, 9679), (9719, 9721), (9767, 9769), (9857, 9859), (9929, 9931) > 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. The math community will not allow me to get research acceptance here any more than with all my other research results. I just like this example to pop people's delusions about that community. I have piles of research results now. 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? I defined mathematical proof. I discovered tautological spaces. I gave the best general method for simplifying binary quadratic diophantine equations. And I get insulted for my efforts. Why should I bother doing anything else with my research at all? Your opinion of me is as irrelevant as your ignorance of the mathematics I understand. You will never understand the best mathematics because you do not want to know it. I prefer instead to toss a result out there every once in a while to remind you how primitive you are, and oh yeah, to explain for people who don't understand how modern math society is on white collar welfare. You need bogus math to pay the bills. I don't. I enjoy knowing. You don't care to know the truth. James Harris
From: junoexpress on 31 Jan 2010 23:53 On Jan 31, 11:29 pm, JSH <jst...(a)gmail.com> wrote: > > Wouldn't matter. There's too much research money in this area and the > twin primes conjecture is too famous. > The math community will not allow me to get research acceptance here > any more than with all my other research results. > Must be terrible being so impotent, unable to prove anything the way a *real* mathematician could. > I just like this example to pop people's delusions about that > community. > Ha! If that's your logic, then by your own standards you're a total failure since more people are being convinced you're a crank from these ng rants and nobody's being convinced you're right. Statements like this show that you are unable to think clearly. > I have piles of research results now. Piles is the correct term. > > I defined mathematical proof. I discovered tautological spaces. So funny how you never produce one though. Guess talk is cheap; you just can't put up the goods though. Snip all the self-pitying delusional rationalizations. HTH, M
From: Joshua Cranmer on 31 Jan 2010 23:58
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. >> 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. > 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. > 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? > 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). > 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? > 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. -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth |