Posted this to Prof. Tao’s blog 3 Nov 11:

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Goodness even the corrections need corrections:

Where we include (P1 – 2) = 1…

The idea there is simply to keep 1 as a placeholder (instead of (2 – 2 = 0) for when we group the individual product terms. It’s not a necessary element, just a convenience to make the elaboration of the infinite product easier to manage and look at.

I see there are still some other minor typos in the preceding posts: a (2 x 3) instead of (2 x 5), and at least one Pn instead of the correct P(n-1)#. Sorry for those.

I think I’ve just about got the correct explanation for Tn, will try to elaborate after the end of the work day. Meanwhile, I should probably explain one thing I’ve glossed over that may have some readers stuck or rejecting my logic.

In the preceding posts I’ve made no specific mention of “actual primes,” but have rather focussed on a discussion of confirmed “non-primes” (as ruled out in any given sieving by a given Pn) and “possible primes” that remain after each sieving step. I’ve suggested that these values can be extrapolated to infinity to obtain the ultimate densities of primes and twin primes.

But what about the pesky ACTUAL primes that develop as we go, namely 2, 3, 5, …. Pn? These actual primes occupy positions I have called “non-primes.” Is this a fatal flaw in my logic? I’m fairly confident this has no bearing, and here is why…

If we take Nn and Tn to be the number of possible primes and possible twins within the nth primorial, where “possible” implies “still standing as candidates after sieving by primes to Pn,” then these values represent the numbers of possibilities within all primorials after the first one (accepting for the moment that Tn remains to be developed/defended).

Within the first primorial, we know that we have n ACTUAL primes, which are sitting on n “non-prime” positions. If we want to be precise, then, we should perhaps state that the number of “possible primes, “within the first Pn# only, is actually Nn + n, and thereafter remains Nn. We may also have some number, which must be less than n, of actual twin primes.

It is trivial to show to show that (Nn + n)/Pn#, the density of primes within the first primorial, converges to Nn/Pn# as n grows. It is also trivial to show that this addition of n “possible primes” within the first primorial has no bearing on the density of “possible primes” in the set of natural numbers after the nth sieving step, when we recall that the first primorial containing the extra n “possible primes” is only one of an infinite number, with the rest all containing exactly Nn “possible primes.”

Hence the density of “possible primes” within the set of natural numbers after n sieving steps remains Nn/Pn#, and the density of twin pairs Tn/Pn#.

Now if only we could prove the definition of Tn, we might have the problem licked. 🙂

(or I could be completely wrong, and given the fact that I’ve never done anything elegant in math on my own, this latter outcome seems somehow more likely, yet this continues to be fun and intellectually stimulating to me, so thanks again for the continued indulgence)

Hi Pete,

Pleased to see you are still pondering this stuff. I’ve also come back to thinking about this problem recently, from much the same angle. In January I got all excited that I had proved the twin prime conjecture only to realise I was being a bit silly, but I still enjoy playing with these ideas, and I think a lot of what you are saying is spot-on, with a few reservations as follows. I will put some more up later as I have more thoughts on this, but here are a couple of observations:

“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?”

I don’t have a formal algebraic answer, but I can give an explanation in words. Each time a primorial length pattern is reiterated you get repeating gaps (for instance when you introduce 7 to the sieve, you get a pattern of 7*30 , with one of the repeating gaps being 29, 59, 89, 119, 149, 179, 209). Within the prime pattern, the next prime p in the sieve will only remove 1/p of each particular gap (or 2/p for twins) – so 7 only sieves out 119 from this particular set of gaps (and 119/121 and 89/91 from the possible twins). This could probably be formally shown by representing the numbers in modulo form, eg, the series above is the start of an infinite series of numbers that are 29 (modulo 30), of which only 1 in 7 can be divisible by 7.

I’d also like to show you a spreadsheet I have that gives a good visual representation of why 2/p is sieved at each step in the twin prime pattern. I’ll try to put it up on my blog and link to it at some point.

“But what about the pesky ACTUAL primes that develop as we go, namely 2, 3, 5, …. Pn? These actual primes occupy positions I have called “non-primes.” Is this a fatal flaw in my logic? I’m fairly confident this has no bearing, and here is why…”

I was pondering this the other day and I’m afraid I do think it is a fatal problem. Firstly, it’s worth looking at Sierpinski carpets for a similar example of fractal geometry. The thing I notice is that the density of a Sierpinski carpet does actually reach zero at infinity. The same thing is true of both the density of prime numbers and twin primes *if* you don’t account for some of the “non-primes” being those pesky “actual primes.” This pattern of repeated iterations of p/p(n) -1 or p/pn-2 would indeed eventually eliminate all possible primes and all possible twin primes. To visualise this, imagine the stream of repeated symmetries going on to infinity. This would eventually cover every number, because no matter how high a prime or twin prime you think of, it would be the first element in a primorial-length pattern of “non-primes.”

So the question must be whether there will always be a twin prime at the *start* of a higher symmetry, not whether the symmetries go on leaving gaps indefinitely, which they can’t. At infinity, all the non-prime gaps are filled. The pesky remainder of actual primes and twin primes is in the end the target, not the density problem.

An interesting thing to think about in this respect is the pattern of semi-primes, numbers with only two factors. It is trivial to show that there is an infinite number of semi-primes, but the interesting question is whether there is an infinite number of semi-primes of the form 6nsquared – 1. (This is (6n+1)*(6n-1), and is therefore the product of any pair of twin primes). The semi-prime pattern is also a fractal pattern, created by iterations of prime eliminations in the Sieve of Eratosthenes.

“Hence the density of “possible primes” within the set of natural numbers after n sieving steps remains Nn/Pn#, and the density of twin pairs Tn/Pn#.”

This is an interesting approach, but I think it falls into the same fallacy I fell into last time round – it doesn’t matter if the density of primes or twin primes that haven’t yet been sieved never falls to zero. It remains a possibility (albeit a slightly absurd one) that at some point the higher primes will somehow conspire to eliminate all higher twin prime candidates. I think we’ve both understood why the twin primes probably do in fact go on forever, because of the way these fractal patterns operate. But I don’t think Tn/Pn# will yield an actual formal proof of this. I might be wrong though.

Thanks for the comment Hugh, I will take some time to absorb it.

I’ve just about got the Tn formula worked out, but don’t want to put it up here until I’m sure it’s right.

I realize the density of twin primes goes to zero with infinity, but this is the same as many fractals, including the Sierpinski carpet, Cantor dusts and the primes – they have infinite components but zero density.

I recognize that the chances of me having actually solved this thing are likely slim to nil, but I feel pretty good about this and want to get it all laid out to see if it holds up.

Cheers,

Pete

Another way of putting the same point. Tn isn’t actually the density of twin primes. It is the density of numbers above Pn# which are still twin prime candidates when the sieve reaches Pn. The actual density of twin primes up to Pn# would be defined by the number of “non-prime” pairs that turned out to be “actual twin primes” once the sieve has reached the square root of Pn#. So the pesky “non-primes” aren’t an error term that can be ignored, they are the actual things we are looking for.

Does that make sense? Will look out for your further enumeration either way.

No I realize Tn isn’t the density of twin primes (twin prime candidates actually), but I contend that Tn/Pn# is the density of twin prime candidates within Pn#, and by extension is also the density of twin prime candidates within the set of natural numbers, and that as n approaches infinity this becomes, by definition, the density of twin primes within the natural numbers, zero.

Hi Pete,

I’ve put up a series of posts on this starting here.that might be of interest:

http://barkerhugh.blogspot.com/2011/11/composite-6n16m-1-grid.html

Cheers,

Hugh

Thanks for the note Hugh, sorry I haven’t looked just yet, but I will when I find some time this week for sure.