I haven’t quite managed to demonstrate the formula for Tn yet, and have been sidetracked looking at prime k-tuples.

We can make the following observations from numerical inspection by stepwise sieving with consecutive primes, from 2 to Pn, that within repetitions of Pn# we find the following:

Where we define:

Nn = number of candidate primes within repetitions of Pn#

Tn = number of candidate twin prime pairs within repetitions of Pn#

Rn = number of candidate prime triplets within repetitions of Pn#, where a prime triplet is a set of three consecutive primes with a maximum separation of 6

Qn = number of candidate prime quadruplets within repetitions of Pn#, where a prime quadruplet is a set of four consecutive primes with a maximum separation of 8

Cn = number of candidate cousin prime pairs within repetitions of Pn#, meaning prime pairs spaced by 4

Sn = number of candidate sexy prime pairs within repetitions of Pn#, meaning prime pairs spaced by 6

QTn = number of candidate prime quintuplets within repetitions of Pn#, where a prime quintuplet is a set of five consecutive primes with a maximum separation of 12

And “candidate” implies that these numbers or groups of numbers remain possible until further sieving by higher primes.

These results were obtained by numerical experimentation in a spreadsheet, using step-wise sieving of the primes from 2 to 17, for all natural numbers to a little past 510510.

By inspection, the values of Nn, Tn, Rn, Qn, Cn, Sn and QTn take the following form:

These results lead to some interesting observations. We can show that there will be a number of non-primes immediately preceding and following every repetition of each primorial, Pn#, equal to at least Pn (with the sole possibility of a prime either immediately prior or following the multiple of the primorial).

It is also very interesting that these formulae all take the same form, with infinite product series of the primes minus the number of primes within the k-tuple.

We can show that the density of twin primes within the set of primes is at least as great as the density of primes within the set of natural numbers, therefore the set of twin primes is infinite. Consider the following:

D(N:Nat) = density of primes within the set of natural numbers

D(T:Nat) = density of twin prime pairs within the set of natural numbers

D(T:N) = density of twin prime pairs within the set of prime numbers

D(R:T) = density of prime triplets in relation to the set of twin prime pairs

D(Q:R) = density of prime quadruplets in relation to the set of prime triplets

D(QT:Q) = density of prime quintuplets in relation to the set of prime quadruplets

From these results, we see that the density of twin primes within the set of primes is 1.32 times the density of primes within the set of natural numbers, hence the set of twin primes is infinite.

By similar arguments, since the density of prime triplets in relation to the set of twin primes is a positive, real multiple of the density of twins within the set of primes, the set of prime triplets is also infinite.

By extension, using the same logic, the sets of prime quadruplets and quintuplets are also infinite.

(I realize there is a significant probability an error exists somewhere in the preceding logic, but I have a fairly good feeling… :-))