This post presents a succinct overview of the key findings obtained from examination of the patterns left within the n-th primorial following n stages of step-wise sieving of primes 2 to Pn. Interesting patterns are observed that seem to demonstrate the infinitude of twin prime pairs and all other possible prime k-tuples.

Conjecture #1: 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.

Individual primes are 1-tuples. Twin primes, cousin twin primes, sexy twin primes and other specific pairs of primes are 2-tuples. Prime triplets, quadruplets and quintuplets are 3-, 4- and 5-tuplets respectively.

Conjecture #2: for a given k-tuple, the density of k-tuples within the set of natural numbers is:

This value is always higher than the true density, but converges (slowly) to the correct value as n approaches infinity.

Conjecture #3: These product series and calculated densities can be used to demonstrate that the set of any arbitrary size k-tuple is infinite, provided more than 1 of such k-tuples exist.

Conjecture #1 can be demonstrated, but not proven, through numerical experimentation. 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

6n = number of candidate prime 6-tuplets within repetitions of Pn#, where a prime 6-tuplet is a set of six consecutive primes with a maximum separation of 16

7n = number of candidate prime 7-tuplets within repetitions of Pn#, where a prime 7-tuplet is a set of seven consecutive primes with a maximum separation of 20

And “candidate” implies that these numbers or k-tuples remain possible until further sieving by higher primes following Pn.

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, QTn, 6n and 7n in the preceding table take the following form:

Conjecture #2 is demonstrated as follows:

After n sieving steps, the number of candidate primes within the n-th primorial is Nn, hence the density of possible primes within the primorial is Nn/Pn#. With only the first n primes used in the sieving process, this pattern within the n-th primorial projects out to infinity unaltered, so the overall density of candidate primes within the set of natural numbers is Nn/Pn#. All of the primes are revealed only when n increases to infinity, hence the density of primes within the set of natural numbers is the limit, with n approaching infinity, of Nn/Pn#.

There is a minor complication that is easily addressed. Within the first repetition of the n-th primorial, the first n primes occupy positions that are non-primes in all subsequent repetitions of the primorial. Therefore the total number of candidate primes needs also include this growing set of actual primes. This number, n, which grows to infinity, actually has a diminishing effect of the density of prime candidates within the set of natural numbers, since n/Nn goes to zero as n increases. This reduces the impact of n on the estimated density as n grows. Further, the effect of this number of actual primes can be made negligible by repeating the primorial an arbitrarily large number of times.

The density of the other k-tuples within the set of natural numbers can be determined by the same logic.

Conjecture #3 is demonstrated as follows:

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:

DN:Nat = density of primes within the set of natural numbers

DT:Nat = density of twin prime pairs within the set of natural numbers

DT:N = density of twin prime pairs within the set of prime numbers

DR:T = density of prime triplets in relation to the set of twin prime pairs

DQ:R = density of prime quadruplets in relation to the set of prime triplets

DQT:Q = density of prime quintuplets in relation to the set of prime quadruplets

Similar densities can be derived for the 6- and 7-tuples.

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. We can show that the set of any k-tuple is infinite, assuming such k-tuples exist, by demonstrating progressively that each subordinate k-tuple also forms an infinite set.

I think the basic logic of the above seems correct for “remaining k-tuple candidates.”

However I see a problem with this bit:

“This value is always higher than the true density, but converges (slowly) to the correct value as n approaches infinity.”

As n approaches infinity, all these equations will tend to zero, surely. Because at infinity, the patterns cover every finite number. You are looking at the primorial-length repetitions (eg 211-420), rather than the first primorial in the pattern (1-210) in order to ignore the error term of the numbers identified within the first primorial pattern as composite which are in fact prime. But in fact, there is no repeating pattern, only one endless infinite pattern which doesn’t repeat. In this pattern, you can’t ignore the primes wrongly identified as composite.

Now, the error term seems quite small at low numbers since there are only a few primes being wrongly identified as composite. But as we approach infinity, the error term approaches infinity, as it closes in on “the number of primes”.

So while this approach clearly does give us a good estimate of the number of remaining k-tuple candidates as you go up through the sieve, in the end it suggests that the density of k-tuples is “zero, with an error term of the density of k-tuples”.

(Not meaning to pick holes, the same problem affects most of what I have worked out about this subject myself, and I think the extra k-tuple stuff here is fascinating in spite of this problem).

Thanks for the comment. I’ll admit that bit bothers me a bit, but I’ve tried to explain it away a couple of times, with the logic being summarized a couple of posts back as:

“There is a minor complication that is easily addressed. Within the first repetition of the n-th primorial, the first n primes occupy positions that are non-primes in all subsequent repetitions of the primorial. Therefore the total number of candidate primes needs also include this growing set of actual primes. This number, n, which grows to infinity, actually has a diminishing effect of the density of prime candidates within the set of natural numbers, since n/Nn goes to zero as n increases. This reduces the impact of n on the estimated density as n grows. Further, the effect of this number of actual primes can be made negligible by repeating the primorial an arbitrarily large number of times.”

I think if I were trying to claim that the NUMBER of primes were converged upon, I’d be in trouble, but since I’m only claiming somethin about the density of primes, I think this error term disappears. Ya think? 🙂

Sometimes I think the same as you say above, but increasingly I think that it is a real problem with this approach. You never actually get an estimate of k-tuple density, only an estimate of candidates in subsequent repeating primorials. Then at the end, the sum doesn’t tend towards a figure that gives you the actual k-tuple density either because it tends to zero. And as a result, it can’t tell you (for instance) whether there is an infinite number of twin primes or a finite number. The equation would give the same result either way, tending to zero with an error term of the finite/infinite number of k-tuples.

Another way of putting it is that an equation that measured the density of primes at any point would need to be able to measure the number of confirmed primes under a certain bound, not the number of unconfirmed primes over that bound.

So I’d say this way of thinking does intuitively back up the idea that there almost certainly is an infinite number of twin primes and helps to show why this is probably true – but it can’t take us any closer to proving it.

There’s a different kind of sieve I’ve been thinking about which might or might not help, will try to post about that later.

Thanks again, I’ve tried to elaborate by writing the idea out mathematically in a new short post today.

I realize the density goes to zero, for the primes, twins and k-tuples. But that doesn’t mean their counts stop growing at some point. We know from a simple proof the primes are infinite. My basic logic is that since we can show the ratio of density of twins within primes is greater than the density of primes within natural numbers, by extension that set (and the sets of all k-tuples by similar logic) must also be infinite. To me this is the softest part of the logic. It feels right to me, but I’m not completely sure it’s an indisputably valid claim.

Cheers,

Pete

Hi Pete,

I realize the density goes to zero, for the primes, twins and k-tuples. But that doesn’t mean their counts stop growing at some point.

No, but nor does it help us to show that the counts don’t stop growing at some point. We know the primes don’t for other reasons, and we suspect the twins don’t but still can’t prove the latter.

“We know from a simple proof the primes are infinite. My basic logic is that since we can show the ratio of density of twins within primes is greater than the density of primes within natural numbers, by extension that set (and the sets of all k-tuples by similar logic) must also be infinite.”

But since your equations tend to zero, they can’t distinguish whether the endpoint is a ratio between a tiny density of twins and a tiny density of primes, or zero density of twins and a tiny density of primes (both ratios would tend to zero). You can prove 2=3 if you allow divisions by zero into the equation, after all.

I meant to put that first para in quotes above…