I continue to find errors. I’ve just corrected the values I’ve claimed a couple of the ratios of densities converge to. The formulae still converge, so these minor errors don’t affect the substance of these findings, but they sure can be distracting!
It should be evident to anyone following this story that I’m treating the internet as a big white board, throwing draft ideas up to see if they stick, and learning from what I can see must be wrong.
Anyway, I think I’ve stumbled across the elusive solution to the neat pattern that’s been evolving. In the last post I claimed, based on fairly compelling (to me anyway) numerical evidence, that within the n-th primorial, Pn#, after sieving by sequential primes from 2 to Pn, there will remain a predictable number of k-tuples:
Where Kn is the number of candidate prime k-tuples within the n-th primorial, Pn#, k is the size of the k-tuple (number of primes within the k-tuple), the constants c1 through cm, and the value of m, depend on the specific configuration of the k-tuple. I’ve further suggested that the density of k-tuple candidates after n sieving steps is Kn/Pn#, and this converges to the actual density of k-tuples in the set of natural numbers as n goes to infinity.
I think I’ve got it. Let’s try the following logic.
Let’s start with single primes. In step 1, we strip away all multiples of 2, leaving 1/2 of all natural numbers as candidate primes in the first instance.
Next we strip away all remaining multiples of 3 from within the set of candidate primes left after the first sieving step. Precisely 1/3 of the prior candidates will be multiples of 3, hence we remove 1/3 of 1/2, or 1/6, leaving 1/3, or 2/6 of all natural numbers as candidate primes after removal of all multiples of 3.
Next we strip away multiples of 5, or remove 1/5 of 1/3, or 1/15, leaving 4/15, or 8/30.
With the next step, we strip away 1/7 of 8/30, or 8/210, to leave 48/210.
Then we strip away 1/11 of 48/210, or 48/2310 to leave 480/2310.
The sequence is clear, at any given n-th stage of sieving we are left with a density of candidate primes within the set of natural numbers equal to the product for i = 1 to n of (Pi-1)/Pi.
This clearly demonstrates the product series formula for Nn.
And what of the twin primes? We follow precisely the same logic, except that for every surviving candidate twin pair after n sieving steps, there are twice as many multiples of the next prime within the set of candidate twin pairs.
After the first sieving step, we are again left with all the odd numbers, each being a potential candidate twin prime, so 1/2 of the set of natural numbers are candidate twins.
When we strip away all multiples of 3, we remove 2/3 of 1/2, or 2/6, to leave 1/6 of all natural numbers as candidate twins after 2 sieving stages.
When we strip away multiples of 5, we remove 2/5 of 1/6, or 2/30, to leave 3/30 as candidate twins.
When we strip away multiples of 7, we remove 2/7 of 3/30, or 6/210, to leave 15/210.
Again the pattern is clear, at any given n-th stage of sieving we are left with a density of candidate twin prime pairs, D_T, within the set of natural numbers equal to the product for i = 2 to n of (Pi-2)/Pi.
Note that in this case we have a first term equal to 1/2 associated with the first stage of sieving. We need to introduce such terms for Pn < (k-1) where k is the size of the k-tuple.
The same logic would necessarily apply to any size k-tuple. For example, in the case of 5-tuples, we would obtain a progression that takes a form of D_5n = D_5(n-1)* (1-(5/Pn)), where D_5n is the density of candidate 5-tuples within the set of natural numbers after n sieving steps, and converges on the actual density of 5-tuples as n approaches infinity.
Development of the first few terms in the product series requires more careful consideration until the size of the primorial under consideration is greater than k.
The density of different configurations of k-tuples of the same size may vary if there are different possible permutations of the same k-tuple. For example, sexy primes being twin primes separated by 6 can either include a third prime between to form a possible triple, or not contain a third prime. Hence the density of sexy primes is greater than that of cousin primes, which cannot contain an interior prime, and therefore have only one permutation, and have the same density as regular twin primes separated by 2. If sexy primes were defined as twins spaced by 6 with NO interior prime, the density would be lower.
Similarly, triple primes can have two forms: Pn, Pn+2, Pn+6; or Pn, Pn+4, Pn+6. The density of all prime triples will be double that of either specific permutation on its own.
The preceding analysis defends the claimed formulae in previous posts for Nn, Tn, Rn, … Kn. Those formulae were a critical component in an argument that seems to demonstrate the infinitude of twin primes and all k-tuple primes.
There are likely still some errors embedded in the logic somewhere, but this feels like it is getting ever closer… 🙂