JSH: Twin primes
in [Math]
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From: master1729 on 1 Feb 2010 08:09 MichaelW wrote : > 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. thanks ! do you mind if i would use that in a paper ? guess not. > > 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". as i expected. > > 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? what do you mean by that ?? > > Regards, Michael W. regards tommy1729
From: JSH on 1 Feb 2010 19:51 On Feb 1, 8:16 am, William Hughes <wpihug...(a)hotmail.com> wrote: > 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? Yeah. It IS a very strong result. I have the prime residue axiom! Also there is a lot of numerical evidence built up by mathematicians over the years. There is a "gotcha" in this result. > (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. Axiom's are considered self-evident. They are necessary because not everything can be proven based on other things, so people accept that some things just must be accepted as being true. Given my prime residue axiom the proof follows easily enough. But there is also a TREMENDOUS amount of numerical verification already done. > (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 The twin primes probability result is such an overwhelming one as mathematicians have been working for years building up data in support of it. Just look in current literature on twin primes for: ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*...*(1/2) It's a very funny situation. Kind of ironic, don't you think? James Harris
From: William Hughes on 1 Feb 2010 20:02 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. Therefore GC is true. - William Hughes
From: JSH on 1 Feb 2010 20:16 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. So there is a refutation by mathematical proof. > Therefore GC is true. > > - William Hughes Wrong. Goldbach's Conjecture is false. Sorry. Everything goes back to the prime residue axiom. The proof of the twin primes theorem relies on the prime residue axiom. That axiom is supported by YEARS of data carefully gathered by mathematicians researching prime numbers. Curious readers can find the equation that results from the prime residue axiom in current literature: ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*...*(1/2) The HUGE gotcha in this thing is the irony of full support from established research and it reveals why it's so hard in a situation when there are people who hate the truth. Real math students would find the support wonderful and amazing. After all, what's true is true. Fighting it is like getting mad because the earth isn't the center of the universe or isn't flat. But as readers see the fight I want them to pay attention to it. If math people can ignore this result then they get continued FUNDING in an area where their own research shows that continued investigation is a waste of time. And the world goes without most knowing that random is about prime preference--or lack of it. Freaking prime numbers may be the reason for random in our physical world. Wow. James Harris
From: William Hughes on 1 Feb 2010 20:32
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 |