Twin Primes Question

[PeterJ]

Hello again folks.

 

Is the following of any interest if it can be proved?

 

P: Relative to any finite set of primes there are infinitely many pairs of consecutive twin primes.

 

Note 1. In case it's the wrong word - by 'relative' I mean that none of the primes in the set are factors of the pair of twin primes.

 

Note 2. This is nothing like a proof of the TPC.

 

Thanks for any replies.

 

 

 

 

 

[Acme]

Hello again folks.

 

Is the following of any interest if it can be proved?

 

P: Relative to any finite set of primes there are infinitely many pairs of consecutive twin primes.

 

Note 1. In case it's the wrong word - by 'relative' I mean that none of the primes in the set are factors of the pair of twin primes.

 

Note 2. This is nothing like a proof of the TPC.

 

Thanks for any replies.

Factors of the pair of twin primes? I don't understand that. By definition Primes have no factors.

[PeterJ]

None of the primes in the set are factors of the two aforementioned adjacent pairs of numbers (n, n+2).

 

.

[Acme]

None of the primes in the set are factors of the two aforementioned adjacent pairs of numbers (n, n+2).

 

.

I still don't understand. Can you give some examples?

[PeterJ]

Okay, but I'm not sure where the problem is yet. Say the set is (2,3,5,7,11,13). The number 289 is not prime, but it is prime relative to this set.

 

Perhaps I could put it differently.

 

P: No finite set of primes produces sufficient products to prevent the occurrence of an infinite qty of consecutive pairs of twin primes.

 

Note that this is not very exciting. It is obvious that no finite set of primes P could prevent the occurrence of a twin prime at P!. I just wondered if by extending a proof to cover consecutive pairs of twins things became more interesting.

Edited April 3, 2014 by PeterJ

[Acme]

Okay, but I'm not sure where the problem is. Say the set is (2,3,5,7,11,13). The number 289 is not prime, but it is prime relative to this set.

The problem was my not understanding what you wrote. I see nothing special about the case you give; I'm sure there is no limit to similar cases.

[PeterJ]

The case I gave is was intended merely as an explanation of 'relative primality'. There are , of course, an infinity of examples.

[Acme]

The case I gave is was intended merely as an explanation of 'relative primality'. There are , of course, an infinity of examples.

OK. So there's no issue and nothing to prove.

[PeterJ]

Yes. There is my proposition to prove. If it's interesting enough to make it worth bothering.

 

I think maybe you should read the proposition carefully. Are you suggesting that you can prove it? I think you'll have a struggle, but if you succeed you'll have answered my question.

[Acme]

Yes. There is my proposition to prove. If it's interesting enough to make it worth bothering.

 

I think maybe you should read the proposition carefully. Are you suggesting that you can prove it? I think you'll have a struggle, but if you succeed you'll have answered my question.

Again, I don't see anything to prove, so my only struggle is trying to understand your point.

[PeterJ]

As nobody else has commented I can't be quite sure that I've put myself clearly. Maybe not.

 

Take a large but finite set of primes, the primes up to 10^50, say. Is this enough primes to ensure that there is eventually a highest consecutive pair of twin primes?

 

Bear in mind that the products of the primes in the set are the only numbers that can have an effect on the answer to this question. If these products are always insufficient to prevent consecutive twin primes from occurring, then it at least possible that there are an infinite qty of such consecutive pairs.

 

I have no idea whether a proof of this possibility would be trivial or interesting, so I thought this was a good place to ask. .

[Acme]

As nobody else has commented I can't be quite sure that I've put myself clearly. Maybe not.

 

Take a large but finite set of primes, the primes up to 10^50, say. Is this enough primes to ensure that there is eventually a highest consecutive pair of twin primes?

 

Bear in mind that the products of the primes in the set are the only numbers that can have an effect on the answer to this question. If these products are always insufficient to prevent consecutive twin primes from occurring, then it at least possible that there are an infinite qty of such consecutive pairs.

 

I have no idea whether a proof of this possibility would be trivial or interesting, so I thought this was a good place to ask. .

...

P: Relative to any finite set of primes there are infinitely many pairs of consecutive twin primes.

...

So, in your first post what you say is a contradiction because there is no infinite set of any natural numbers within a finite set of natural numbers.

 

As to your rephrasing, since the twin prime conjecture is unproven I see no basis for proving something similar for consecutive twin primes. [Note: Consecutive (a pair of) twin primes can be referred to as 'prime twins' or 'prime pair'.] While it is one thing to say how likely a prime character is on an interval, it is quite another to say what interval or where on it the prime character may appear.

 

Perhaps my comments will draw comments from the others. (I don't know how likely that is though. )

[PeterJ]

Sorry. I have no idea what you're saying here.

 

What has your point about 'no infinite set of natural numbers within a set of natural numbers' got to do with anything?

 

You say, 'While it is one thing to say how likely a prime character is on an interval, it is quite another to say what interval or where on it the prime character may appear'.. This may be so, but I have no idea what this sentence means or why it's relevant.

 

I'm asking a very simple question. A yes or no answer would be fine. Would a proof be interesting or trivial, it's a multiple choice question with just two possible answers.

Edited April 3, 2014 by PeterJ

[Acme]

 

So, in your first post what you say is a contradiction because there is no infinite set of any natural numbers within a finite set of natural numbers.

Sorry. I have no idea what you're saying here.

 

You said:

... Relative to any finite set of primes there are infinitely many pairs of consecutive twin primes.

So how can there be infinitely many anything in a finite set?

 

...

I'm asking a very simple question. A yes or no answer would be fine. Would a proof be interesting or trivial, it's a multiple choice question with just two possible answers.

No.

[John]

If we have some finite set S of primes, then take the product of those primes. Call this product n. Now consider kn where k is some positive integer. Since each prime in S divides n, each prime in S must also divide kn. Now, it's a fact that consecutive integers are always coprime. Thus, for any kn, we have that neither kn - 1 nor kn + 1 is a multiple of any prime in S, and so they must form a pair of "primes relative to the set." Since there are infinitely many positive integers, it follows that there must be infinitely many such pairs.

[PeterJ]

Thank you John

 

Your example seems to refer only to single twins. I'm speaking about adjacent pairs of twins. Is this not something quite different, and a little more difficult to prove?

[John]

My understanding was you were asking for pairs of integers (n, n + 2) such that no prime in our set was a factor of n or n + 2. Are you asking for something else?

Edited April 4, 2014 by John

[PeterJ]

Yes!

 

I'm talking about consecutive instances of twin primes. Two twin primes in a row, the longest sequence there can ever be.

 

The TPC asks whether there are infinitely many single twin primes. I thought it might be slightly interesting to show that it is at least possible, no more than that, that there are infinitely many consecutive twin primes. Or, if you like, that it would be impossible to prove that there is a highest instance of two twin primes in a row.

 

I didn't mean to make a big deal about it. It may have been proved a thousand times for all I know, or it may be completely useless.

Edited April 4, 2014 by PeterJ

[John]

I think there may be some miscommunication going on here.

A twin prime is a prime with another prime two units away from it. The prime and its twin are referred to as a twin prime pair. So when you say "two twin primes," I'm taking you to mean a single twin prime pair, but you may be meaning something else.

 

Since every third odd number greater than three is divisible by three, the closest two twin prime pairs can be to each other is three units apart. So when we talk about "prime quadruplets," for example, we mean four primes of the form p, p + 2, p + 6, and p + 8. It's an open question whether there are infinitely many sets of prime quadruplets, and if there are, then this immediately implies that the twin prime conjecture is true. The proof of any similar statement involving the infinitude of twin prime pairs with some property also implies that the twin prime conjecture is true.

In your OP, my understanding is that you asked for a proof of the infinitude of twin prime pairs relative to a given set of primes, i.e. pairs of numbers n, n + 2 such that for any prime p in our original set, p does not divide n and p does not divide n + 2. This is what my earlier proof was about.

You seem to be asking for something else now, but I'm still not sure what.

[PeterJ]

Okay. Perhaps I should have said 'prime quadruplets'.

 

My thought is this. On the face if it, it is possible that there is only a finite quantity of prime quadruplets. This is regardless of the truth of the TPC. It could well be that the first 10,000 primes are enough to ensure that there is a highest quadruplet. A proof that it would not be possible to prove that this is the case seemed like it might be useful, or would at least be a curiosity. But maybe not. I just wondered.

 

Am I be misreading your proof above? It appears to relevant only to single twin primes, and to more or less restate Euclid's argument.

Edited April 4, 2014 by PeterJ

[Acme]

I think there may be some miscommunication going on here.

 

A twin prime is a prime with another prime two units away from it. The prime and its twin are referred to as a twin prime pair. So when you say "two twin primes," I'm taking you to mean a single twin prime pair, but you may be meaning something else.

 

...

You seem to be asking for something else now, but I'm still not sure what.

He is asking about 'prime twins', which is as he said, a pair of twin primes with no interceding primes.

...Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. ...

Twin primes @ Wiki >> http://en.wikipedia.org/wiki/Twin_prime

 

So, for example, [11:13|17:19] is a 'prime twin' or a 'prime pair'.

 

Yes!

 

I'm talking about consecutive instances of twin primes. Two twin primes in a row, the longest sequence there can ever be.

 

The TPC asks whether there are infinitely many single twin primes. I thought it might be slightly interesting to show that it is at least possible, no more than that, that there are infinitely many consecutive twin primes. Or, if you like, that it would be impossible to prove that there is a highest instance of two twin primes in a row.

 

I didn't mean to make a big deal about it. It may have been proved a thousand times for all I know, or it may be completely useless.

Now that I understand what you are asking, I do find it interesting. I'll see if I can find a list of your quarry at OEIS.

Edited April 4, 2014 by Acme

[PeterJ]

Great stuff! Thanks for hanging on in there.

[Acme]

Great stuff! Thanks for hanging on in there.

You're welcome. Thanks for hanging in with me as well. I can be a bit crusty; or so I have been told.

 

So I started searching OEIS for the term 'prime pair' and it returned 175 pages of listings. So far I have got through about the first 20 and while it's chock full of interesting goodies I haven't found the listing. Or at least the listing as I expect to see it. Anyway, I'll give you the link and maybe you can start at the end and work your way back.

 

prime pair @ OEIS >> https://oeis.org/search?q=prime%20pair

 

Not exactly what we want, but here's a listing of the gaps between twin prime pairs.

 

>> https://oeis.org/A167132

...Let G_n denote the twins gap between two consecutive twins, thus a twins gap is the difference between two consecutive twins (p_n, p_n+2) and (p_m, p_m+2), i.e. the difference between p_m and p_n+2. G_n = p_m - (p_n +2) We have: G_1 = 0, G_2 = 4, G_3 = 4, G_4 = 10;

 

0, 4, 4, 10, 10, 16, 10, 28, 4, 28, 10, 28, 10, 4, 28, 10, 28, 10, 28, 34, 70, 10, 28, 58, 46, 28,...

Edited April 4, 2014 by Acme

[phyti]

Eliminating 2 and 3, all primes are of the form 6x-1 or 6x+1, i.e. twin primes. The twin prime conjecture has not been proven yet. If there are gaps in the twin primes, that implies bigger gaps. It would seem that solving the tpc is more fundamental than searching for other intervals, 4, 6, etc.

 

[PeterJ]

Acme- Unfortunately I'm not able to understand the quote you give at the end of your post. But I will go have a look at OEIS link. I had no idea such places existed.

 

phyti - I think maybe you need to read the thread again.