JSH: Prime gap equation, 3 years grace
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From: Mark Murray on 24 Jul 2010 12:26 On 24/07/2010 16:32, JSH wrote: > And I think mathematicians routinely lie about even their most > sacrosanct areas of research, like especially with prime numbers. An opinion you are most welcome to. > Progress? .... is something that will happen in spite of anything you attempt. You are irrelevant to it. Keep posting, though. Its entertaining. M -- Mark "No Nickname" Murray Notable nebbish, extreme generalist.
From: MichaelW on 24 Jul 2010 19:40 On Jul 25, 1:19 am, JSH <jst...(a)gmail.com> wrote: > > Interesting. What if you shifted to 32? In the same interval? How > close is the count then? See below. > > And if you're curious you MIGHT then consider what my prime gap > equation says for both. I'll admit I am not doing any actual math in > this area as the very title of this post is about my 3 years grace. > > But hey, I can't stop others, eh? If you look at the equation then it creates two values. The first is a probability of a pair occurring at a particular prime when the gap is a power of 2. The second is the Corr value which adjusts for the prime factors of the gap. The product of the two gives the final probability. The result is that for any particular interval (say, 10^6 to 10^7) there is a number J that will be the predicted number of pairs for a gap of 2,4,8,16,32 etc; this J is the sum of the probabilities over the primes in the interval. The count for 6,12,18,24,36 etc (gaps with only 2 and 3 as factors) will be 2*J since Corr=2 in this case. The count for other gaps can be discovered by calculating Corr and multiplying by J. The need for the Corr multiplier comes out of the maths and has been known at least since the 19th century. If you see this link http://en.wikipedia.org/wiki/De_Polignac%27s_conjecture then the same calculation can be see in the discussion of C_n. The hard part is to determine J. I used this formula: Pi_2(x) = 2*C_2*x/ln(x)^2 where C_2 is the twin prime constant. Better is the Li_2 integral but I could not find a calculator for this integral online. > > To really thoroughly check though, you'd check for EVERY prime gap > possible within that interval, and that is what I've meant above. > And my point has been that you don't need to check for every prime gap as once you have J you have all the values. Note that this is a direct conclusion from the prime gap equation that you posted so I am agreeing with your maths regarding this point. > Computers don't care. They'll do the work without screaming at you. > > The tables are around so every single prime gap possible from 10^6 to > 10^7 can be checked against my prime gap equation, or for any other > interval for which a table is available. THAT check isn't amenable to > rationalizations, and the standard theory collapses under it, with > horrible results in comparison (I hope, eh?). Could you provide a link for a table of prime gaps? Google has not been my friend. Regarding standard theory; we have three processes for calculating J: (1) Pi_2(x) given above (2) Li_2(x) which is the integral in the Wiki link above (3) Your probability function You seem to be implying that (3) is somehow superior to (1) and (2). I will agree that subject to the need for a constant multiplier (your equivalent of C_2) that (3) gives a much better answer than (1). Regarding (2) I have a conjecture: Equation (3) gives a close approximation of (2); they are in fact essentially the same. > > That type of check is what can yank checks out of professors hands and > have them scrambling to find new ways to make money to feed their > kids--as the government dole which I call white collar welfare goes > away--pay their mortgages and keep their wives in decent clothes. > > James Harris If my conjecture is correct (and I have proved it to my satisfaction) then in fact your prime gap equation reinforces standard theory. For reference Li_2(10^7) - Li_2(10^6) is 50506. The prime gap equation before constant multiplier gives 56537. The actual number of pairs for g=2 is 50811, for g=32 is 50369 and for g=18 is 101012 (giving a J of 50506). Regards, Michael W.
From: JSH on 24 Jul 2010 20:31 On Jul 24, 4:40 pm, MichaelW <ms...(a)tpg.com.au> wrote: > On Jul 25, 1:19 am, JSH <jst...(a)gmail.com> wrote: > > > > > Interesting. What if you shifted to 32? In the same interval? How > > close is the count then? > > See below. > > > > > And if you're curious you MIGHT then consider what my prime gap > > equation says for both. I'll admit I am not doing any actual math in > > this area as the very title of this post is about my 3 years grace. > > > But hey, I can't stop others, eh? > > If you look at the equation then it creates two values. The first is a > probability of a pair occurring at a particular prime when the gap is > a power of 2. The second is the Corr value which adjusts for the prime > factors of the gap. The product of the two gives the final > probability. > > The result is that for any particular interval (say, 10^6 to 10^7) > there is a number J that will be the predicted number of pairs for a > gap of 2,4,8,16,32 etc; this J is the sum of the probabilities over > the primes in the interval. > > The count for 6,12,18,24,36 etc (gaps with only 2 and 3 as factors) > will be 2*J since Corr=2 in this case. The count for other gaps can be > discovered by calculating Corr and multiplying by J. > > The need for the Corr multiplier comes out of the maths and has been > known at least since the 19th century. If you see this link > > http://en.wikipedia.org/wiki/De_Polignac%27s_conjecture > > then the same calculation can be see in the discussion of C_n. > > The hard part is to determine J. I used this formula: > > Pi_2(x) = 2*C_2*x/ln(x)^2 > > where C_2 is the twin prime constant. Better is the Li_2 integral but > I could not find a calculator for this integral online. > > > > > To really thoroughly check though, you'd check for EVERY prime gap > > possible within that interval, and that is what I've meant above. > > And my point has been that you don't need to check for every prime gap > as once you have J you have all the values. Note that this is a direct > conclusion from the prime gap equation that you posted so I am > agreeing with your maths regarding this point. > > > Computers don't care. They'll do the work without screaming at you. > > > The tables are around so every single prime gap possible from 10^6 to > > 10^7 can be checked against my prime gap equation, or for any other > > interval for which a table is available. THAT check isn't amenable to > > rationalizations, and the standard theory collapses under it, with > > horrible results in comparison (I hope, eh?). > > Could you provide a link for a table of prime gaps? Google has not > been my friend. Oh, I could be wrong. It's not like I've been looking for such tables. There ARE I know tables of the primes themselves and finding gaps with those should be trivial to program. But you should be able to do a lot down in the lower ranges just by sieving for primes. > Regarding standard theory; we have three processes for calculating J: > > (1) Pi_2(x) given above > (2) Li_2(x) which is the integral in the Wiki link above > (3) Your probability function > > You seem to be implying that (3) is somehow superior to (1) and (2). I > will agree that subject to the need for a constant multiplier (your > equivalent of C_2) that (3) gives a much better answer than (1). Math that gives the same answer should be related. > Regarding (2) I have a conjecture: Equation (3) gives a close > approximation of (2); they are in fact essentially the same. > > > > > That type of check is what can yank checks out of professors hands and > > have them scrambling to find new ways to make money to feed their > > kids--as the government dole which I call white collar welfare goes > > away--pay their mortgages and keep their wives in decent clothes. > > > James Harris > > If my conjecture is correct (and I have proved it to my satisfaction) > then in fact your prime gap equation reinforces standard theory. Then ask yourself: why isn't it already taught as I've presented it? > For reference Li_2(10^7) - Li_2(10^6) is 50506. The prime gap equation > before constant multiplier gives 56537. The actual number of pairs for > g=2 is 50811, for g=32 is 50369 and for g=18 is 101012 (giving a J of > 50506). Interesting. Ok, here's the short spiel as to why what I say is so revolutionary: my position is that probability alone is the answer. If so, then why do further research? The prime residue axiom IS a big deal because if it is an axiom then you can derive the prime gap equation and explain prime gaps. People who want to know why would then have the answer. What did you think all the arguing was over? James Harris
From: MichaelW on 24 Jul 2010 20:48 On Jul 25, 10:31 am, JSH <jst...(a)gmail.com> wrote: > > Then ask yourself: why isn't it already taught as I've presented it? > It is; references have been provided. Problem is that you don't go far enough. For example your prob function rapidly approaches c/ln(p) for some constant c. This makes it an alternative method to using the PNT to count the pairs. Current theory went way beyond this approach 100 years ago. If you mean why not use the algorithm to calculate the number of prime pairs then the answer is that it is computationally inefficient. Speaking as someone who has run the code there are much faster ways of doing what your equation does. Basically the code looks like this: <code> for each odd n in the interval { is the largest prime smaller than sqrt(n) in the probability? If no then the probability is multiplied by p-2/p-1 if n is a prime then add the probability to the running tally } </code> For a decent size interval (say around 10^15) then you would have to go through the iteration 10^15 times and there is some heavy resource use inside the loop. Regards, Michael W.
From: Dri one on 25 Jul 2010 23:52
"Mark Murray" <w.h.oami(a)example.com> wrote in message news:4c4b1410$0$2522$da0feed9(a)news.zen.co.uk... > On 24/07/2010 16:32, JSH wrote: >> And I think mathematicians routinely lie about even their most >> sacrosanct areas of research, like especially with prime numbers. > > An opinion you are most welcome to. > >> Progress? > > ... is something that will happen in spite of anything you attempt. > You are irrelevant to it. > > Keep posting, though. Its entertaining. > see ? JSH is all TROLL. |