How reasonable to assume a prime gap of at least 10 before a pair of Twin Primes?

[Nelphine]

If we assume that the Twin Prime Conjecture is true (and thus there are infinite number of primes that are 2 apart), how reasonable is it to assume that there will be an infinite number of Twin Primes that are preceded by a prime that is at least 10 lower than the first of the Twin Primes? (I actually only need it to be at least 8 lower, but that's not actually possible, so 10 it is.) As clarification, if the preceding prime was 1000 lower than the pair of Twin Primes, that would be fine.

What if I also assume that the K Prime Conjecture holds?

In other words, even assuming that the Twin (or K) Prime Conjecture holds, what can we assume about the distance between triples of primes (as opposed to pairs of primes)

 

[Amir Livne]

My 2 cents on you first question.

I would be very, very surprised if there were an infinite number of prime twins and almost all of them had another prime close to them (say ±10 like in your example).

By the pigeonhole principle, for one of the 10 distances you would have an infinite number of occurences, which means there are an infinite number of prime triples (n, n+D, n+D+2).
This is not unreasonable in itself, but is less likely that infinitude of prime twins.

However, the vast majority of natural numbers are not primes. They generally don't tend to cling together, so I'd expect about the same density of primes around twin pairs as everywhere else. But the conjecture that there is almost always a prime in one of the 10 preceding numbers to a pair means that the density in this area is 0.1, whereas the density in $\mathbb{N}$ is 0.

This is contrary to the usual distribution of primes. For example, you can find distances between consequtive primes that are as big as you like.

 

[Nelphine]

No, what I'm asking is:

Assuming there are an infinite number of Twin Primes, how reasonable is it to assume that NOT all twin primes will be preceded by a twin prime that is less than 10 difference.

So if I assume that p(k) and p(k+1)=p(k)+2 are a pair of twin primes, I want p(k-1)<= p(k)-8

As a specific example, since 2782991 and 2782993 are a pair of twin primes, I want 2782989, 2782987 and 2782985 to not be prime.

Obviously for any given set of twin primes this wouldn't be a reasonable assumption, but for, say, every million pairs of twin primes, would it be reasonable to assume that at least one of them held this to be true?

 

[ramsey2879]

No, what I'm asking is:

Assuming there are an infinite number of Twin Primes, how reasonable is it to assume that NOT all twin primes will be preceded by a twin prime that is less than 10 difference.

So if I assume that p(k) and p(k+1)=p(k)+2 are a pair of twin primes, I want p(k-1)<= p(k)-8

As a specific example, since 2782991 and 2782993 are a pair of twin primes, I want 2782989, 2782987 and 2782985 to not be prime.

Obviously for any given set of twin primes this wouldn't be a reasonable assumption, but for, say, every million pairs of twin primes, would it be reasonable to assume that at least one of them held this to be true?

If p(k) and p(k) +2 are twin primes > 3, I think that since p(k)-2 and p(k)-8 would both be divisible by 3, it would not be unreasonable for large p(k) to assume that on average, p(k) - 4 and p(k) -6 are also not prime.

 

[RamaWolf]

Let (p,q,r) be any three consecutive prime numbers, then we form classes [a,b] defined by

(p,q,r) $\in$ [a,b] if q-p = a and r - q = b

We neglect the classes [1,2] with elemeent (2,3,5) and [2,2] with element (3,5,7)

Now consider [4,2] with elementsw (7,11,13), (13,17,19), (37,41,43), (67,71,73), ...

an further on: (103837,103841,103843), (103963,103967,109969), (104677,104681,104683)

Can we imagine, that this class is finite??

 

[Nelphine]

RamaWolf, I wouldn`t be surprised if that class was infinite. However, that`s exactly what I don`t want. What I`m looking for is to show that the class of classes [a,2] where a > 8 is infinite.

 

[ramsey2879]

RamaWolf, I wouldn`t be surprised if that class was infinite. However, that`s exactly what I don`t want. What I`m looking for is to show that the class of classes [a,2] where a > 8 is infinite.

That would be no easier to prove than to prove the Twin Prime conjecture in the first place, but it is certainly a reasonable assumption if you assume the twin prime conjecture is true.