mersenneforum.org Polignac's Conjecture Question
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 2020-11-16, 23:50 #1 travisfauxhawke   Nov 2020 1002 Posts Polignac's Conjecture Question Are there flaws in any of the considerations below? A) All Prime Numbers greater than 3 are contained within the Sets 6x-1 and 6x+1 for x>0 [1]. B) The Set 6x-1 contains an infinite number of Prime Numbers [2]. C) P_(6x-1) (x), the probability that a given number in the Set 6x-1 is a Prime Number, is: - 1) equivalent to the Prime Number Density, π_(6x-1)(x)/x (by definition) [3] - 2) always greater than zero (because there are an infinite number of Prime Numbers in the Set 6x-1) D) The Set 6x+1 contains an infinite number of Prime Numbers [2]. E) P_(6x+1) (x), the probability that a given number in the set 6x+1 is a Prime Number, is - 1) equivalent to the Prime Number Density, π_(6x+1)(x)/x (by definition) [3] - 2) always greater than zero (because there are an infinite number of Prime Numbers in the Set 6x+1) F) The probability of finding a gap of size g is - 1) P_(g=2) = P_(6x-1) (x) * P_(6x+1) (x) - 2) P_(g=2) = P_(6x-1) (x) * P_(6x+1) (x-1) - 3) P_g is as per Table 1 here when g is 6, 8, 10, 12 ... G) Because P_(6x-1) (x)>0 and P_(6x+1) (x)>0, then P_g (x)>0 for any value of x or g; therefore, there are infinitely many cases of two consecutive prime numbers separated by any given gap. H) Based on the considerations above and the Prime Number Theorem, predictions can be made for Prime Gap Densities for any value of x (as has been done here , where the Actual Prime Gap Densities agrees well predictions).
 2020-11-17, 00:21 #2 VBCurtis     "Curtis" Feb 2005 Riverside, CA 10010011110102 Posts The probabilities you use as inputs tend asymptotically to zero. You treat them as finitely positive, which isn't correct. You also treat the probabilities as independent, but they are not (necessarily). Your argument amounts to "there are probably an infinite number of prime pairs of a particular gap, because the individual probabilities don't vanish to zero". It's not a proof- more of a summary of a heuristic.
 2020-11-17, 03:26 #3 Dr Sardonicus     Feb 2017 Nowhere 10001010111002 Posts A Survey of Results on Primes in Short Intervals indicates that, in order to insure $\pi(X + y) - \pi(X)\; \sim\;\frac{y}{\log(X)}\;(X\;\rightarrow\;\infty)$ in accord with PNT, y has to be a larger function of X than you might hope.
 2020-11-18, 04:36 #5 VBCurtis     "Curtis" Feb 2005 Riverside, CA 2·5·11·43 Posts Well, I think both your numbered assumptions are incorrect. I think the probabilities are *not* independent, and if they're not the rest of your reasoning doesn't achieve anything. This article may help- it shows independence is not a valid assumption between consecutive primes. https://www.quantamagazine.org/mathe...racy-20160313/
 2020-11-18, 22:45 #6 travisfauxhawke   Nov 2020 22 Posts You're right that if assumptions are incorrect, we can't achieve accurate results. So, I suggest that we test the assumptions and see how well the predicted results match actual results. If the assumptions are incorrect, then the agreement between the predictions and results will be poor and your opinion will be validated. Assumption #1: π_(6x-1)(x) = π_(6x+1)(x) Empirically n/log(n) is a lower limit on Prime Counting Function in that it underestimates the number of prime numbers. So,(1) π(n) ≥ n/log(n)and then based on Assumption #1(2) π_(6x-1)(x)/x ≥ 3 / log(6x+1)(3) π_(6x+1)(x)/x ≥ 3 / log(6x+1)Figure 2a and 2b from the last post show empirically that Assumption #1 is pretty accurate. Not only that, but Eq. (2) and (3) are also accurate, but could be improved if we used a more accurate estimator of π(n) than n/log(n) (but we can stick with n/log(n) as there is a proof of PNT using this function). Assumption #2: P_(6x-1)(x) and P_(6x+1)(x) are independent Twin Primes will occur when 6x-1 and 6x+1 are both prime. Using Assumption 2 the probability of finding Twin Primes for a given value of x is the product of P_(6x-1)(x) and P_(6x+1)(x). So, (4) P_g=2 = P_(6x-1)(x) * P_(6x+1)(x) = π_(6x-1)(x)/x * π_(6x+1)(x)/xand substituting (2) and (3) into (4) (5) P_g=2 ≥ 3 / log(6x+1) * 3 / log(6x+1)This can be extended for Cousin Primes, Sexy Primes, etc. as described in the first post and P_g can be evaluated for various values of g and x. Figure 3 from the last post shows that Assumption #2 is also pretty accurate as it shows excellent agreement between predicted and actual values in both trend and values. The empirical evidence doesn't show that Assumptions #1 & #2 result in poor agreement between predicted and actual results. On the contrary, predictions based on these assumptions seem pretty accurate. So, to persist in thinking that the assumptions are incorrect requires some explanation to resolve the paradox as to why the assumptions, despite being incorrect, make accurate predictions. I hope to hear that you have a solution to this paradox. Otherwise, I would suggest that the assumptions need to be treated as correct.
 2020-11-19, 00:14 #7 VBCurtis     "Curtis" Feb 2005 Riverside, CA 127A16 Posts Did you read the article? Or perhaps I'm missing what you are trying to achieve; it appears you're rephrasing long-known heuristics and trying to then claim that you've reached a new conclusion because you phrase them all in terms of primes being 1mod6 or 5mod6. I now fear we're talking right past each other- maybe you just like that the heuristics work nicely, while I'm trying to point out that you are no closer to a demonstration of infinite numbers of twin primes than anyone else who reads the heuristic formula for the number of twin primes below a particular bound. "my heuristics work for these small numbers, see?" doesn't advance a claim. Maybe you haven't looked at big-enough numbers; maybe, like in the article, the correlations between consecutive primes are subtler than you realize.
2020-11-19, 18:09   #8
mart_r

Dec 2008
you know...around...

2·11·29 Posts

Quote:
 Originally Posted by VBCurtis This article may help- it shows independence is not a valid assumption between consecutive primes. https://www.quantamagazine.org/mathe...racy-20160313/
I have to say I'm quite astounded about how big of a deal this discovery seems to be.
If I had done some of those calculations and found out how primes (in base 10) with the same end digits seemed to "repel each other", I would've done some quick modular restriction maths and shaken it off as something rather trivial and not really worth mentioning, and I'm almost dead sure many other people have done so before.
Working out the details of course is a task for those more in the know than me.

Actually I stumbled upon this kind of problem myself a couple of years ago (for my constant y such that floor(p#*y) is always prime), and found a similar solution.

 2020-11-19, 22:22 #9 travisfauxhawke   Nov 2020 48 Posts Yup, read the article. It was interesting. I just like that the heuristics work nicely. Nothing beyond that except to see how closely this method matches reality when the conjecture is formulated this way. If you want to predict the number of "sexy" primes between 0 and 60,001. How would you do it?
 2020-11-19, 22:42 #10 VBCurtis     "Curtis" Feb 2005 Riverside, CA 2·5·11·43 Posts Yep, I misunderstood you. I thought you were claiming you had created something more rigorous than the twin primes conjecture to "prove" the infinitude of twin primes. I don't even know what a sexy prime is, so I would have quite a lot of work to do to enumerate them!
2020-11-20, 20:13   #11
Gelly

May 2020

438 Posts

Quote:
 Originally Posted by VBCurtis Yep, I misunderstood you. I thought you were claiming you had created something more rigorous than the twin primes conjecture to "prove" the infinitude of twin primes. I don't even know what a sexy prime is, so I would have quite a lot of work to do to enumerate them!
It's exactly what you think it is.

(primes with a gap of six: p and p+6 are prime)

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