JSH: Twin primes
in [Math]
|
From: William Hughes on 1 Feb 2010 11:16 On Feb 1, 1:37 am, JSH <jst...(a)gmail.com> wrote: > 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. A rather strong result. How do you know it is true? (You have not even given a form in which it can be proven.) It is not enough to say, no one can show that it is false. No one can show that GC is false but that does not mean GC is proven. (How many legs does a dog have if you call a tail a leg? Calling something an axiom does not make it one) - William Hughes
From: Joshua Cranmer on 1 Feb 2010 12:01 On 02/01/2010 09:42 AM, MichaelW wrote: > This example struck me as rather convenient. For example if the range > is from 113^2 to 116^2 then the prediction is under 11 twins but the > actual count is 7. I suspect James is being selective. Well, of course. No one wants to select the examples that cast the work into the most doubt. > I have written a program and looking at the results between 100 and > 200 the match varies from really good (139^2 to 146^2 gets the correct > answer (30) within 1%) to the really bad (157^2 to 163^2 predicts 27 > but there are only 17). I suspect that most of the "real" results are consistently below the "expected". Is that true? -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
From: MichaelW on 1 Feb 2010 15:23 On Feb 2, 4:01 am, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > On 02/01/2010 09:42 AM, MichaelW wrote: > > > This example struck me as rather convenient. For example if the range > > is from 113^2 to 116^2 then the prediction is under 11 twins but the > > actual count is 7. I suspect James is being selective. > > Well, of course. No one wants to select the examples that cast the work > into the most doubt. > > > I have written a program and looking at the results between 100 and > > 200 the match varies from really good (139^2 to 146^2 gets the correct > > answer (30) within 1%) to the really bad (157^2 to 163^2 predicts 27 > > but there are only 17). > > I suspect that most of the "real" results are consistently below the > "expected". Is that true? > > -- > Beware of bugs in the above code; I have only proved it correct, not > tried it. -- Donald E. Knuth Actually not too much either way. Here's some from 113^2 onward: 113**2 and 114**2 2 twins. Pred = 3.64185994 113**2 and 115**2 5 twins. Pred = 7.43546404 113**2 and 116**2 7 twins. Pred = 10.7738356 113**2 and 117**2 8 twins. Pred = 14.2639514 113**2 and 118**2 15 twins. Pred = 18.6645322 113**2 and 119**2 19 twins. Pred = 21.8511596 113**2 and 120**2 22 twins. Pred = 24.7342987 113**2 and 121**2 27 twins. Pred = 29.1348795 113**2 and 122**2 28 twins. Pred = 33.3837161 113**2 and 123**2 28 twins. Pred = 36.8738319 113**2 and 124**2 33 twins. Pred = 41.4261568 113**2 and 125**2 34 twins. Pred = 45.2197609 113**2 and 126**2 38 twins. Pred = 49.3168533 113**2 and 127**2 42 twins. Pred = 53.7174341 In this case the prediction exceeds the number of actual twins all the way through. On the other hand for 139^2 139**2 and 140**2 5 twins. Pred = 4.85789479 139**2 and 141**2 8 twins. Pred = 8.53811811 139**2 and 142**2 13 twins. Pred = 13.3960129 139**2 and 143**2 16 twins. Pred = 17.6650720 139**2 and 144**2 21 twins. Pred = 21.1980864 139**2 and 145**2 28 twins. Pred = 25.9087722 139**2 and 146**2 30 twins. Pred = 29.7362045 139**2 and 147**2 37 twins. Pred = 34.7413082 139**2 and 148**2 41 twins. Pred = 39.1575762 139**2 and 149**2 45 twins. Pred = 44.0154710 In this case the prediction is pretty close although mostly a little under. My observation is that at high numbers there is a big gap between primes. The prob formula however generates a constant multiplier so if the twin primes cluster or go sparse (which happens a lot) the formula diverges and usually fails to "catch up". There appears to be a relationship between the formula for "prob" and the reciprocal of the zeta function for s=1, at least as p -> infinity. Anyone able to look into this? Regards, Michael W.
From: master1729 on 1 Feb 2010 07:21 > 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 notice about the range 13^2 - 17^2 that if we sum the probabilities of the previous ranges : 2.5 + 2.25 + 4.69 + 2.53 + 5.67 = 17.64 17.64 is about the sum of actual values : 17 !! intresting ! > > 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. the overestimates dont seem large at first. > > 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) that seems convincing not ? > > > 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. > tommy1729
From: master1729 on 1 Feb 2010 08:02
> On 02/01/2010 09:42 AM, MichaelW wrote: > > This example struck me as rather convenient. For > example if the range > > is from 113^2 to 116^2 then the prediction is under > 11 twins but the > > actual count is 7. I suspect James is being > selective. > > Well, of course. No one wants to select the examples > that cast the work > into the most doubt. > > > I have written a program and looking at the results > between 100 and > > 200 the match varies from really good (139^2 to > 146^2 gets the correct > > answer (30) within 1%) to the really bad (157^2 to > 163^2 predicts 27 > > but there are only 17). > > I suspect that most of the "real" results are > consistently below the > "expected". Is that true? > > -- > Beware of bugs in the above code; I have only proved > it correct, not > tried it. -- Donald E. Knuth yes MichaelW , we would like to get more details of your research. 157^2 to 163^2 predicts 27 but there are only 17. notice 26569 - 24649 = 1920. 10 off for an interval of 1920 seems not so bad. on the other hand of course 27/17 ... what if we test the intervals 5^2 .. 7^2 5^2 .. 9^2 5^2 .. 11^2 5^2 .. 13^2 5^2 .. ..^2 maybe that gives better results ?? regards tommy1729 |