Recently wrote a couple of posts on Terry Tao’s blog about twin primes, on 31 October and 1 November. Just copying it all over here so I’ve got a record on here:
More on primes and twin primes from a complete amateur. Looking for help answering a question…
What I’ve tried to describe in previous posts are some patterns that develop within primorials. For the n-th primorial, Pn#, the distribution of primes can obviously be fully developed by repetition of all primes less that sqrt(Pn#). The removal (or sieving) of all non-primes can be divided into two broad components: (1) a symmetric pattern of non-primes removed through repetition of all primes from 2 to Pn (symmetric since these all divide evenly into Pn#), and (2) a non-symmetric distribution of non-primes generated by all other primes from P(n+1) to the highest prime < sqrt(Pn#).
Numerical experimentation shows that the symmetric component of non-primes projects outward, presumably to infinity, for every subsequent repetition of Pn#, so that primes can never fall on those positions, and primes appear, at "random" (clearly not random, but apparently random) intervals at the other positions that were not "non-primes" within Pn# screened by the primes to Pn.
Interestingly, if only to me, is that the number of "possible primes" (i.e. not "non-primes" screened by the primes to Pn), which I will call Nn, within any given primorial, Pn# ( obviously defined as the product of all primes to Pn), is related to the primorial as follows: Nn = (product for all i=1 to n) [Pn – 1]
(sorry I don't have LaTEX)
In my last post I linked to some notes describing results of numerical experimentation that show that similar patterns develop for twin primes. Within any given primorial, all possible twin primes can be sieved via symmetric removal due to primes to Pn, and then non-symmetric removal by higher primes to sqrt(Pn#). The symmetric pattern of "non-twins" removed by, and lingering "possible twins" left alone by, the sieving of primes to Pn, then repeats to infinity for all repetitions of Pn#, so that higher twins will always only occur at these "possible twin" locations.
Even more interesting (probably still only to me, haha) is that the number of "possible twins" within any given primorial is ALSO related to the primorial, similar to the way the "possible primes" are, but with the following form, where Tn is the number of "possible twins" within any given repetition of a primorial:
Tn = (product for all i=2 to n) [Pn – 2]
Alternatively, one can describe the number of "possible twins" removed when going from Pn# to P(n+1)# as:
T(n+1) = P(n+1)*Tn – 2 * (product for all i=2 to n) [P(n-1) – 2]
= P(n+1)*Tn – 2 * Tn
= (Pn – 2)*Tn
Using a couple of random examples to illustrate this observation:
consider P4 = 7, where P4# = 210.
The number of possible positions for primes within repetitions of 210 projecting out to infinity is 48, a result you can check for yourself fairly easily (work out Pi(mod 210) for as many primes as you like and you will find exactly 48 unique values). This is (7-1)*(5-1)*(3-1)*(2-1)
Similarly, the number of possible twin prime pairs within repetitions of 210 is T4=15, which is (7-2)*(5-2)*(3-2). You can readily see that T3 would be 3, or (5-2)*(3-2).
Note that T4 = 15 = (P4 – 2)*T3 = 5*3
I can't work out the math to prove these results, but they seem far to simple to be accidental, and to not continue to infinity. Can anyone suggest a way to demonstrate these results algebraicly, or otherwise?
Thanks for humouring me.
dangit, no editing function… there are a few silly typos that should be obvious by context, like some n’s in products should be i’s
Goodness I left numerous irritating little typos in that post, sorry. To correct a few:
“Possible primes” within the nth primorial, Pn#, is:
Nn = (product for all i=1 to n) [Pi – 1]
“Possible twins” within the nth primorial is:
Tn = (product for all i=2 to n) [Pi – 2]
T(n+1) = [P(n+1) – 2]*Tn
For a few moments, let’s suspend disbelief and pretend these definitions for Nn and Tn are proven.
I now propose that the density of primes, within all natural numbers, is less than Nn/Pn#, and converges on Nn/Pn# as n approaches infinity. I will claim this without proof for the moment and maybe elaborate later. More disbelief to suspend, sorry. 🙂
I also propose that the density of twin pairs, within all natural numbers, is less than Tn/Pn#, and converges on Tn/Pn# as n approaches infinity.
The ratio of the density of twin pairs to the density of primes, within all natural numbers, which is the same as the density of twin pairs within all primes, can be taken as (Tn/Pn#)/(Nn/Pn#) = Tn/Nn, n approaching infinity
Let’s now compare the density of twin pairs within the set of primes to the density of primes within the set of natural numbers, as:
[Tn/Nn]/[Nn/Pn#], as n approaches infinity.
I believe it is fairly straightforward to show that this ratio converges on about 1.320…, the twin prime constant.
This result tells us, I think, that the set of twin primes corresponding to the set of primes is 1.32 times larger than the set of primes corresponding to the set of natural numbers (with apologies, if this isn’t written correctly from a mathematical perspective). Since the set of primes is infinite, hence the set of twin primes is also infinite.
Does that make sense?
I’ve claimed a couple of things in here without proof – the first being the existence and definitions of Nn and Tn, which I’ve demonstrated (to myself anyway) through numerical experimentation but not proven mathematically. I would think that, given the simplicity of the relationships they should be easy to develop algebraically, but admit that they are beyond my ability. The second unproven claim is that the ratios converge to the values indicated as n approaches infinity. I think those claims are much easier to rationalize, and the fact that they lead to the twin prime constant would seem to suggest they must be correct.
Thanks for reading. 🙂