Twin Primes Question

[Acme]

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.

...

I empathize; the entries can be quite cryptic.

 

I'll have a go at 'splainin. So we have their list, 0, 4, 4, 10,...

 

[i think] The first zero means there is no gap/difference between the first prime pair. [3,5] & [5,7]. (5-5=0) The next in their list, a 4, means there is a gap/difference of 4 between the next prime pair. [5,7] & [11,13]. (11-7=4) Then a gap/difference of 4 between the third pair, [11,13] & [17,19]. (17-13=4) Rinse & repeat.

 

No doubt if I have botched that, someone will correctify my air.

[John]

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.

It is possible, as no proof to the contrary has been found. What I'm reading now is you're looking for sets of prime quadruplets relative to a finite set of primes, i.e. four numbers p, p + 2, p + 6, p + 8 such that no prime in our set divides any of these four numbers. Is that correct? Or is it p, p + 2, p + n, p + n + 2 for some integer n, such that there are no relative primes between p + 2 and p + n?

 

My proof above uses similar reasoning to Euclid's proof of the infinitude of primes. What my proof says is that, given any finite set of prime numbers, we can find a twin prime pair relative to the set, i.e. numbers p, p + 2 such that no prime in the set divides either number. Moreover, there are infinitely many such "relative twin prime pairs." This is what I understood you to be asking in your OP.

Edited April 4, 2014 by John

[PeterJ]

Okay Acme. Got it. Thanks. OEIS is doing my head in but I'm going through backwards looking our for anything that sounds relevant.

 

John. It would be the first of your two formulations that I'm concerned with. Two adjacent twin primes each at 6n+/-1. The vital word in my OP was 'consecutive'. Without that a proof would, as you say, be trivial.

 

I didn't know that two primes at p and p+4 could be considered a twin, and perhaps should have made it clear that this is not what I mean here by 'twin prime'.

[John]

But in your OP you didn't seem to be asking about primes, but rather numbers not having any of a given finite set of primes as a factor. Are you wanting to know about actual prime quadruplets now? Or just prime quadruplets relative to the original set of primes? The latter won't necessarily be of the form 6n +/- 1.

 

And no, two primes p and p + 4 are not considered twin primes.

 

It might help if you restate precisely what you're wanting proved.

Edited April 5, 2014 by John

[PeterJ]

I'm talking about consecutive twin primes. That is to say, a twin prime at 6n+/- followed immediately by another one. I don't know how to improve on this explanation. The TPs at 6+/-1 and 12+/-1 would be an example.

 

I'm asking if it is would be interesting if we could prove that no amount of primes on a list, (whether they are sequential or not), would be sufficient to generate enough multiples to prevent there being an infinitude of consecutive twin primes. (NOT consecutive primes, but consecutive twin primes.)

 

The quadruplet thing seemed relevant because I noticed (or thought I did) that OEIS includes data on pairs of primes at p and p+4 under the twin primes section. So I wanted to make it clear that this is not what I'm talking about.

Edited April 5, 2014 by PeterJ

[John]

Since the product of any number of primes is by definition not prime, I don't see how generating multiples from a list of primes would have any effect on the infinitude or lack thereof of consecutive twin prime pairs.

[PeterJ]

Sorry John, I don't see your objection.

Edited April 6, 2014 by PeterJ

[Unity+]

Sorry John, I don't see your objection.

He is saying that he doesn't see, and I don't either, how generating numbers from multiplication of primes shows that there are infinite twin primes.

[PeterJ]

Urgh. Going through the OEIS dbase is painful. I've checked 30 pages without finding anything relevant. Trouble is that even if it is relevant I may miss it, due to the technical language used. I'll go back later and do more checking.

He is saying that he doesn't see, and I don't either, how generating numbers from multiplication of primes shows that there are infinite twin primes.

No, nor me.

 

I wish I could express myself more clearly, but surely what I've written already is enough to make it clear that I am not suggesting there is a proof for an infinity of twin primes. I mean really, a person is hardly likely to come on a maths forum and ask if a proof of the TPC would be interesting.

[Acme]

Urgh. Going through the OEIS dbase is painful. I've checked 30 pages without finding anything relevant. Trouble is that even if it is relevant I may miss it, due to the technical language used. I'll go back later and do more checking.

 

No, nor me.

 

I wish I could express myself more clearly, but surely what I've written already is enough to make it clear that I am not suggesting there is a proof for an infinity of twin primes. I mean really, a person is hardly likely to come on a maths forum and ask if a proof of the TPC would be interesting.

Painful is an understatement; it's downright torture. Anyway, I think I understand what you're after in spite of the communication, so I'll try & jump that gap. By all means correct me if I screw it up. Back in post #11 you said:

... Is this enough primes to ensure that there is eventually a highest consecutive pair of twin primes? ...

From this I gather the issue is really the gap between primes, or more to your point the gap between prime pairs. Since such a gap must be all composite numbers, this is why you have talked about relative primality. Oui/no? So I think your issue is actually a special case of the prime gap 'problem'.

 

Anyway, whether I hit the nail or mashed my thumb, you may find this article of interest.

 

Bounded Gaps Between Primes >> http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html

 

Whether we grow up to become category theorists or applied mathematicians, one thing that I suspect unites us all is that we were once enchanted by prime numbers. It comes as no surprise then that a seminar given yesterday afternoon at Harvard by Yitang Zhang of the University of New Hampshire reporting on his new paper Bounded gaps between primes attracted a diverse audience. I dont believe the paper is publicly available yet, but word on the street is that the referees at the Annals say it all checks out.

What follows is a summary of his presentation. Any errors should be ascribed to the ignorance of the transcriber (a category theorist, not an analytic number theorist) rather than to the author or his talk, which was lovely.

...

By the way, in 2004, Daniel Goldston, János Pintz and Cem Yıldırım were able to show that there are infinitely many pairs of primes at most 16 apart… if something called the Elliott–Halberstam conjecture is true.

Edited April 6, 2014 by Acme

[John]

That's the clarification I'm asking for: are we talking about consecutive pairs of actual twin primes, or are we talking about consecutive pairs of twin "primes relative to the set," i.e. consecutive pairs of numbers not having one of the primes in our original set as a factor?

 

If the former, then I don't see multiples of the primes in our initial set having any impact one way or the other. If the latter, then we can immediately simplify the question as hinted at before.

[PeterJ]

Painful is an understatement; it's downright torture. Anyway, I think I understand what you're after in spite of the communication, so I'll try & jump that gap. By all means correct me if I screw it up. Back in post #11 you said:

From this I gather the issue is really the gap between primes, or more to your point the gap between prime pairs. Since such a gap must be all composite numbers, this is why you have talked about relative primality. Oui/no? So I think your issue is actually a special case of the prime gap 'problem'.

 

Anyway, whether I hit the nail or mashed my thumb, you may find this article of interest.

 

Bounded Gaps Between Primes >> http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html

 

My main problem with the OEIS list is that I don't speak the languge. Even that article is beyond me. Where it starts, 'A *prime gap* is an integer ...' I have no idea what the formula that follows means. As far as I can understand it though it doesn't seem to help.

 

Yes, I do think this seems to be a special case of the prime gap problem. I thought it was a simple little idea, but it seems not. I'll say more about it in my reply to John.

That's the clarification I'm asking for: are we talking about consecutive pairs of actual twin primes, or are we talking about consecutive pairs of twin "primes relative to the set," i.e. consecutive pairs of numbers not having one of the primes in our original set as a factor?

 

If the former, then I don't see multiples of the primes in our initial set having any impact one way or the other. If the latter, then we can immediately simplify the question as hinted at before.

That's it. We're talking about consecutive pairs of twin primes relative to the set, i.e. consecutive pairs of numbers not having one of the primes in the original set as a factor.

 

It's a trivial thing really, but I find it intriguing. The proof may not even work, but at this point I;m just trying to see what it would mean if it did.

 

This would be nothing like Euclid's proof. It would not prove that any particular case of n,n+2 is a twin prime. It would state only that insofar as it is possible to predict the density of primes from the general rules governing the behaviour of the products of the primes, (which is the only determinant of the density), the possibility of two consecutive twin primes occuring infinitely many time can never be ruled out.

 

NB. It would NOT state that there is an infinite quantity of pairs of consecutive twin primes, just that if there is not we'll never know it.

 

I suppose it is a proof that the claim of the TPC cannot be refuted, but there must be many ways to prove this.

Edited April 6, 2014 by PeterJ

[John]

My reading of what you're asking now is whether we can show that the statement "there are only finitely many sets of consecutive twin prime pairs," or even the statement that "there are only finitely many twin prime pairs," is not provable in whatever axiom system (I guess ZF or ZFC). This is not trivial (as far as I can see anyway).

Edited April 6, 2014 by John

[PeterJ]

Now we seem to be on the same wavelength. I hope it's not trivial, but I'm still uncertain. If it is not, I'll have a go at formalising my casual attempt.

 

It can definitely be shown that there is only one instance of three twin primes in a row, and that this is the longest sequence possible, but I've been presuming that this would not be news.

[imatfaal]

Now we seem to be on the same wavelength. I hope it's not trivial, but I'm still uncertain. If it is not, I'll have a go at formalising my casual attempt.

 

It can definitely be shown that there is only one instance of three twin primes in a row, and that this is the longest sequence possible, but I've been presuming that this would not be news.

 

 

Put some number on things Peter and other members may get a better grip of what you are driving at

[PeterJ]

Oh no! I'm not going to start getting into the details. Certainly not the numbers.

 

All I'm concerned with here is the question of whether a proof as described would be old hat, trivial, interesting or useless. If it is not trivial then I may move on to say some more, but there's no need to open another can of worms at this point.

 

I suppose I could restate the question in case it helps.

 

If we could prove that no finite quantity of primes would produce sufficient products to ensure that there is a highest twin prime pair (two consecutive twin primes), then would this be of any use to anyone?

Edited April 7, 2014 by PeterJ

[John]

I suppose it could be useful. However, I think we'd ultimately end up just talking about actual consecutive twin prime pairs.

Without being very rigorous, the fact that all primes greater than 3 are equal to 1 or 5 mod 6 means any product involving any subset of such primes will also be equal to 1 or 5 mod 6. Clearly we have plenty of gaps between multiples in which to find our relative twin prime pairs in this case. In particular, any set of four successive even integers will constitute a pair of twin prime pairs relative to the set of primes greater than 3.

Adding just 3 to this set means any product of a subset will be equal to 1, 3 or 5 mod 6 (not 0, since any multiple of 3 landing on a multiple of 6 must be an even multiple, and we're not including 2 in our initial set of primes yet). Again, any set of four successive even integers gives us what we want.

 

Similarly, adding just 2 to the set means any product of a subset will be equal to 1, 2, 4 or 5 mod 6. We're still able to find consecutive twin prime pairs relative to this set. If our product n is even, then we still have that n - 1 and n + 1 are prime relative to our set. Each of these will be odd, and the closest any two odd "composites relative to the set" can be in this case is 6, i.e. if c is an odd composite relative to the set, then the closest possible odd composites relative to the set will be c - 6 and c + 6. There are a few cases to try here, but I'm fairly certain we can find consecutive relative twin prime pairs for each case. Similar reasoning applies if our product n is odd in the first place.

Finally, adding 2 and 3 both to our set ultimately just yields the question of consecutive twin prime pairs in the first place.

Of course, there's always the chance I've made some error in reasoning or arithmetic that yields the above invalid.

Edited April 8, 2014 by John

[PeterJ]

Sorry John. I don't understand your objection.

 

What does your argument prove? I can't discern this.

Edited April 8, 2014 by PeterJ

[John]

It's not an objection. I'm just saying that finding infinitely consecutive twin prime pairs relative to a finite set of primes seems easily doable, and the question doesn't appear to become difficult until we start with the set of all primes, which yields something related to the actual twin prime conjecture.

 

What I showed above is that given any set of primes greater than 3 (really greater than 2), we can immediately find consecutive twin prime pairs relative to the set. Similarly if we start with 2 and all primes greater than 3. It's only when we include 2 and 3 that things get interesting, and at that point (since all primes greater than 3 are similar in the sense of being one of two numbers mod 6) we're basically talking about the set of all primes.

Edited April 8, 2014 by John

[PeterJ]

You may be right about this John. If so my question is answered. The problem is that as yet I can't see quite what you're saying

 

I think I should have specified the set of primes S as being all primes below R. Iow, S would include 2 and 3.

 

How would you go about showing that relative to S (as newly defined) there are always two consecutive twins? If you can do this I'm sorted.

[John]

In post #42, I actually made a mistake in my paragraph about adding just 2 to a set of primes greater than 3. I split the cases where our product is even and where it is odd, but I defined n earlier in the thread to be the product of all the primes in the set, so the product will definitely be even. The rest of the reasoning still applies, though, as far as I can tell.

As for a finite set of primes including both 2 and 3, my point is that I'm not sure this changes the question much versus just talking about the entire set of primes, and I don't know (though admittedly I haven't thought about it a whole lot) how to go about finding successive twin prime pairs. Including 2 and 3 already gives us two of the major properties regarding consecutive twin prime pairs, namely that no even integer is prime and every third odd after 3 is not prime.

 

If we look at the set containing only 2 and 3, then we already see these restrictions in practice. The first prime relative to the set that isn't already a prime regardless is 25, making the first set of successive relative twin prime pairs {(23, 25), (29, 31)}. I don't immediately see an easy way to guarantee that we can always find successive pairs of twin primes relative to a set containing 2, 3 and other primes, but perhaps someone better versed in number theory could easily do it.

Edited April 9, 2014 by John

[Acme]

...

If we look at the set containing only 2 and 3, then we already see these restrictions in practice. The first prime relative to the set that isn't already a prime regardless is 25, making the first set of successive relative twin prime pairs {(23, 25), (29, 31)}. ...

Hi John

Either the bolded is a typo or I don't understand your reasoning. Since 25 is not prime it can't be part of a twin prime or a prime pair. ??

[John]

We're talking here about primes relative to our initial set. If the initial set is {2, 3}, then 25 is such a relative prime, since neither 2 nor 3 divides 25.

[phyti]

Oh no! I'm not going to start getting into the details. Certainly not the numbers.

 

All I'm concerned with here is the question of whether a proof as described would be old hat, trivial, interesting or useless. If it is not trivial then I may move on to say some more, but there's no need to open another can of worms at this point.

 

I suppose I could restate the question in case it helps.

 

If we could prove that no finite quantity of primes would produce sufficient products to ensure that there is a highest twin prime pair (two consecutive twin primes), then would this be of any use to anyone?

If you prove there are only a finite number of twin primes, then it implies a finite number of consecutive multiple prime pairs, 4, 6, etc. It seems you are just restating the tpc in a different way.

Quartets depend on twins, but the reverse is not true.

[PeterJ]

phyti - No, I'm definitely not restating the TPC. And I cannot prove that there are a finite quantity of twin primes. In fact I would expect there to be an infinite qty. It is, however, possible to show that there is no reason to suppose there is a finite quantity, and that it is possible that they are infinite, and the question here is whether it would be of any use to show this.

 

John - We seem to have finally sorted our misunderstandings. The question now is, as you say, can someone with a better knowledge of number theory do it? I really do not know.

 

I've emailed an academic who was helpful to me a similar question a few years back, but no reply as yet.