Talk:Twin prime conjecture
|Text from Twin prime conjecture was copied or moved into Twin prime with this edit. The former page's history now serves to provide attribution for that content in the latter page, and it must not be deleted so long as the latter page exists.|
- 1 Removed text
- 2 Du's proof
- 3 anon external links
- 4 Merge with Twin prime?
- 5 A valid proof of the twin prime conjecture or just another conjecture?
- 6 Impossible Statement
- 7 Goldbach
- 8 Corrected twin primes constant
- 9 Significance?
- 10 Wiener-Ikehara theorem
- 11 My Conjecture
- 12 Claim to be from Euclid
- 13 History
I removed the following passage:
- It's unknown if Gödel's incompleteness theorem has something to do in this conjecture. If the first statement of TPC is undecidable,it means no computer and no human can ever found either there is a biggest pair or it's endless.However, the problem does have an answer even if that's the case.
This text doesn't seem to have any clear meaning (at least, not clear and correct at the same time). --Trovatore 05:16, 2 March 2006 (UTC)
I removed this version too:
- The statement there are an infinite number of primes p such that p + 2 is also primecould be an unprovable statement.If it is, that means no computer and no human can ever know either there is a biggest pair or it's endless.
Let me explain in more detail what's wrong with it:
- There's no such thing as "an unprovable statement" full stop. (For example, any statement is provable if you add it as an axiom. Of course you may wind up proving everything in that case, but you will make the original statement provable.)
- So something needs to be said about the formal theory in which TPC might be unprovable. Is it Peano arithmetic? Is it ZFC? What about adding large cardinal axioms?
- But even if the text were changed to reflect the above, the stuff about no one ever being able to know whether TPC is true simply does not follow. Perhaps there are ways of knowing it that go beyond the formal theory from which TPC was shown unprovable.
- (Of course, this still leaves out the technical possibility that TPC might be proven false from a theory we believe in, such as one of those listed in the second bullet point. Personally I just don't see that happening, but still, if something were added about independence/undecidability, the language probably shouldn't exclude the possibility outright.) --Trovatore 15:29, 2 March 2006 (UTC)
Are you saying that answering to the TPC question is simply very difficult? Some of the books I've read insist that knowing the answer could be impossible. (such as David Deutsch's "The Fabric of Reality" and Satoy's "The Music of Primes".)WAREL 16:36, 2 March 2006 (UTC)
- I'm saying that simply showing that TPC is independent of some fixed formal theory, say PA or ZFC, is not sufficient to show that the answer is unknowable. It's conceivable, I suppose, that it is unknowable, but no techniques currently exist for proving that something is unknowable (and it's hard to imagine what such a technique could look like). --Trovatore 16:42, 2 March 2006 (UTC)
So, you're saying that we definitely could "know"(in general sense) the answer to the TPC sometime.Correct?WAREL 16:53, 2 March 2006 (UTC)
- "Definitely could" is a tricky concept. I'm saying that no independence result, of the sort we currently understand, can rule out the possibility of knowing the answer. That doesn't necessarily mean we ever will know the answer, just that the techniques you seem to be talking about are not strong enough to rule out the possibility that we will. --Trovatore 17:44, 2 March 2006 (UTC)
Is it possible that this question is determined as we will not ever know the answer?WAREL 20:05, 2 March 2006 (UTC)
- I'm reluctant to say it's impossible, but I don't see how we could come to know that we will never know it, short of God showing up and telling us we'll never know. --Trovatore 20:07, 2 March 2006 (UTC)
So,the answer is "it's possible". Isn't it? And also, it's possible that someone will show that we could never know the answer.Isn't it?WAREL 20:20, 2 March 2006 (UTC)
- A very slippery word, "possible". I do not claim to know that no one will ever show we can't know the answer. That does not, in my estimation, justify making the claim that such a situation is "possible"; I think that would be severely misleading. --Trovatore 20:23, 2 March 2006 (UTC)
Why does it not justify making the claim that such a situation is "possible"?WAREL 01:55, 3 March 2006 (UTC)
- Because it sounds like a positive affirmation, when it's actually nothing but an admission that I don't have a complete theory of knowledge, which is what I'd need (or something close to it, anyway) to completely rule out future knowledge that we'll never know the answer. --Trovatore 03:07, 3 March 2006 (UTC)
I have a question. You said it's possible that this question is determined that we will never know the answer. Why is that?WAREL 07:47, 6 March 2006 (UTC)
- No, I did not say that. You keep trying to get me to say that. --Trovatore 14:48, 6 March 2006 (UTC)
Do you mean we're just too stupid yet to know the answer?WAREL 17:05, 6 March 2006 (UTC)
Does anyone have any information of someone talking about Du's proof?I wonder if it's valid or not.WAREL 07:08, 3 March 2006 (UTC)
- (With due respect to the authors of that paper.) If we listed every time someone claimed to have proved the twin prime conjecture, this would be an awfully long and uninformative article. I am removing the reference to that proof, as the arxiv does not count as a peer-reviewed publication. Dmharvey 13:25, 3 March 2006 (UTC)
Is is not reviewed? How did you know that? WAREL 14:43, 3 March 2006 (UTC)
- (1) I can't find anything on Mathematical Reviews Online to suggest that it has been published. (2) The arxiv, by definition, is not peer-reviewed, except for a brutal first pass which culls out any obvious nonsense. See Arxiv. (3) Also read peer review, especially the section entitled Criticisms of peer review. Dmharvey 14:59, 3 March 2006 (UTC)
What do you rely on when a mathematician comes up with a new result? I wrote in wikipedia perfect number section recently that odd perfect number N has at least 9 distinct prime factors, and at least 12 if 3 does not divide N (Nielsen 2006). I knew it from Arxiv.But I hear no one complaining about this result.Is it because it's trustable?How could you tell untrustable results from trustable results?WAREL 17:44, 3 March 2006 (UTC)
- This is a very good question, and I can't give you any clear-cut answer. I saw your edits to perfect number and was tempted to revert, but did not, mostly because I don't know anything about work in that area, and I didn't have the energy to chase up any references you gave. I didn't even realise that your source was the arxiv. (You might consider reverting those edits yourself :-)) The situation with twin primes is quite special. The twin prime conjecture is one of those very famous statements which gets proved every second week. Therefore I set the bar much higher, and certainly the arxiv is not good enough. If someone very famous claimed to have proved the twin prime conjecture and had a pre-print on the arxiv, and if the mathematical world was buzzing with talk about it, then perhaps I would consider mentioning a rumour, but even then it's probably not good form. The thing about the arxiv is that it's not very difficult to have things posted there. Pretty much anyone can upload whatever they like;
there are moderators who glance over it briefly to make sure it's not pornography or random characters, and then there it is. One of the main reasons for its existence is to help people establish that they were the first ones to write something down. I don't even trust things I read in journals — but it's the best we've got as a community, and it seems to be the level of trust that Wikipedia generally considers acceptable. You might want to reconsider your approach to how much you trust things you read on the arXiv. After all, you don't trust things you read on Wikipedia, do you? :-) Dmharvey 18:15, 3 March 2006 (UTC)
So,what you're essentially saying is that Du is not famous enough ,right? But the time someone who's very famous says "I read it through ,and it's correct." is when you trust.I'm not blaming your attitude,though. I's just curious about what your "community" was. WAREL 18:31, 3 March 2006 (UTC)
- Well, it's true that I have never heard of Du, but that doesn't mean that much, as you seem to be suggesting. As for my community, well I am a graduate student in number theory, I spend several tens of hours each week hanging around a mathematics department, surrounded by people who live and breathe mathematics, earn their daily living from mathematics, and some of whom are reasonably famous in their fields. This micro-community is part of a much larger macro-community of mathematicians around the world. Again, maybe that doesn't mean much to you, although it does mean a lot to me. I should also point out that there are different levels of trust. For example, I've carefully considered Euclid's proof that there are infinitely many prime numbers, and I believe it, and I don't think I need anyone any more to trust; I don't even need to trust Euclid himself any more. I also happen to believe that Fermat's last theorem is true, basically because the proof has been accepted by the community that I am part of,
even though I don't understand the proof completely myself. But I don't trust it as much as I trust Euclid's proof. My point is that in Wikipedia, as it's a collaborative project, we don't have the luxury of writing down whatever we personally believe in. There is a broad consensus that academic journals are generally good enough (although I'm sure that gets debated from time to time), but that things like the arXiv are probably not. It's possible that other editors (such as yourself) disagree with me about this last sentence. Dmharvey 18:49, 3 March 2006 (UTC)
Is someone even reading Du's thesis to see if it's true? If no one is reading just because he's not famous, it's not fair to say his proof is probably wrong,is it? WAREL 22:12, 3 March 2006 (UTC)
- What about the proof outlined here? Or this one? Or this one? Or this one? And this person claims to have proved it, without any details provided. Have you checked all of those proofs? It's not fair to say their proofs are probably wrong, is it? Are you aware of this proof, which originally appeared on the arXiv, and recently (2004) generated a lot of quite serious mathematical interest.... but turned out to be wrong? Dmharvey 22:36, 3 March 2006 (UTC)
So, are you saying that Du's paper is not worth reading? If so,is it unworth for you or the entire mathematical community? WAREL 22:45, 3 March 2006 (UTC)
- No, I'm not saying that it's not worth reading. It may well be. If I hear from people I trust that it's worth reading and might be correct, then I may well read it. But otherwise I'm not planning to read it, because reading things (especially the kind of reading one must do to verify a proof) is time-consuming, and I have lots of other things to do (like have interesting conversations on wikipedia :-)). What I'm trying to say is that I don't currently have any evidence that anyone has read it and checked it, let alone that anyone whose mathematical skills I trust has read it and checked it. If it appeared in a professional journal, that would be some indication that such a thing has happened. It wouldn't mean that the proof is correct — sometimes even journals publish things that turn out to be wrong — but it would sure increase my interest. Dmharvey 22:51, 3 March 2006 (UTC)
Are all people you trust saying that Du's paper is unworth reading?WAREL 23:01, 3 March 2006 (UTC)
- I haven't asked any of them. Dmharvey 23:01, 3 March 2006 (UTC)
So I'll wait to write about it until then.WAREL 23:05, 3 March 2006 (UTC)
- Indeed, the job of Wikipedians is not to proofread any paper which shows up at ArXive. Our job is to include only well-established and published material on which there is consensus that it is correct. WAREL, in case you did not read that, you may want to take a look at Wikipedia's policy about No original research. Oleg Alexandrov (talk) 23:32, 3 March 2006 (UTC)
No, I just asked if there "is" a consensus.WAREL 00:01, 4 March 2006 (UTC)
Is it alright with everybody to have some sentences of Du's result once one of the people who Dmharvey trust said it's worth reading?WAREL 04:16, 4 March 2006 (UTC)
- That's fine with me. As soon as someone I trust tells me it's worth reading, I'll let you know. By the way WAREL, your English is pretty good. Some problems with grammar sometimes, but not bad. Where did you learn English? Dmharvey 12:42, 4 March 2006 (UTC)
- That would be a nice conclusion to this fruitful discussion, were it not for the fact that WAREL keeps inserting the link to Du's paper elsewhere . I will thus share my findings.
- The paper is not worth reading, and in fact, it is not worth the paper on which I printed it. Leaving aside the insanely obfuscated notation which the authors set up, the proof of the main lemma 4.5 is invalid, it is a meaningless manipulation with symbols. The nonsense actually begins in definition 2.3 and the ensuing discussion leading to eq. 2.19, but until lemma 4.5 it does not bite. The rest of the paper seems to be more or less correct, but it only consists of a few elementary properties of dubious value, and more to the point, it does not get us any closer to a proof of the twin prime conjecture. -- EJ 01:54, 5 March 2006 (UTC)
In fact, it looks like they also proved Goldbach's conjecture, one week earlier: . What's fascinating about these two papers is how incredibly similar they are; the lemma numbering matches up beautifully. Dmharvey 02:51, 5 March 2006 (UTC)
- LOL! I can't wait when they will extend the method to cover Riemann Hypothesis as well. -- EJ 03:26, 5 March 2006 (UTC)
Look carefully,though. They didn't say they'd proved Goldbach's theorem.WAREL 05:13, 5 March 2006 (UTC)
- Okay, now I have looked carefully. The abstract says "We proved that any even number not less than 6 can be expressed as the sum of two old primes, 2n=p_i+p_j." I don't know what an "old prime" is, but seeing as the only other time the word "old" appears in the article is in the word "Goldbach" and "oldest unsolved problem", I think it's safe to say that it's a typographical error, and is supposed to be "odd". Then, theorem 1.1 (the main "theorem" of the paper) is stated as follows: "There is at least one pair of double primes for any even number . Again, I don't know what a "pair of double primes is", but I take it to mean a pair of primes numbers such that ." This is in fact what is suggested by their statement that "[The goldbach conjectures states that] for any even number 2n there exists a pair of double primes such that . Additionally, in the example for n = 23 given just
before definition 2.2, they state that there are four double prime pairs, namely (3, 43), (5, 41), (17, 29), (23, 23), which agrees with this meaning. I'm guessing that the strange English grammar is a literal translation of some Chinese grammar that hasn't worked too smoothly. I don't know enough Chinese to say. So yes, it certainly looks to me like they are claiming they have proved the Goldbach conjecture. Dmharvey 12:24, 5 March 2006 (UTC)
- One could also point out that the statement and proof of corollary 6.2 is almost free of such grammar errors, and makes clear what they are up to. -- EJ 17:14, 5 March 2006 (UTC)
WAREL, I can't help but notice that you still did not provide any evidence that Du's proof is notable, let alone accepted. You agreed above that you will wait with such additions until people who Dmharvey trusts say that it's worth reading. Can I ask why don't you keep your word? - EJ 17:14, 5 March 2006 (UTC)
Their English is indeed terrible. Have you found any mathematical mistakes that make the tpc theorem not working?WAREL 20:30, 5 March 2006 (UTC)
- Did you read my post above? Once again: everything from definition 2.3 till the end of section 2 is complete nonsense. Given a set of integers of size t, there is no way of computing the number of its elements which are not congruent to 0 or -2 modulo p, knowing t and p alone. It can be as low as 0, and it can be as large as t. -- EJ 21:08, 5 March 2006 (UTC)
An anon using a Rogers account keeps pushing links to the work of Martin C. Miner, owner of a Canadian-based real estate website, relating to probabilistic arguments in favor of TPC. Now, I'm in favor of mentioning such arguments; it's my understanding that there are excellent heuristic reasons to believe TPC is true, if not quite as obvious as the ones for Goldbach's conjecture (the latter you can practically work out in your head). But I don't see any reason to think the best arguments along this line come from this non-peer-reviewed website. Moreover I'm suspicious that the anon may be Miner.
It would be good if someone could write something sourceable about the probabilistic arguments. --Trovatore 18:46, 11 March 2006 (UTC)
- Well, someone already did. We have a whole section on the Hardy-Littlewood conjecture. -- EJ 19:20, 11 March 2006 (UTC)
- Ah, I see now. I should have read the article more carefully. --Trovatore 19:27, 11 March 2006 (UTC)
The name is Winer and this is Winer speaking now. One of my students was trying to place my articles on wikipedia after he noticed they were removed. He was shocked at the arrogance he encountered in trying to replace them. Quite frankly, so am I. These are 'external links' not legal testimony. If this is how mathematicians work, then there's no wonder you're all stuck in the stone ages. —The preceding unsigned comment was added by 220.127.116.11 (talk • contribs) 18:40, 13 March 2006 (UTC)
- Wikipedia's position on original research is clear. Unless your proofs (or commentary on those proofs) are published in a peer-reviewed journal, they don't exist for the purpose of Wikipedia. This is not necessarily how mathematicians work -- although a quick study of one of the links shows you've defined "random" sufficiently differently than the standard definition that one would suspect other terms being redefined as well -- but it's how a non-expert-based encyclopedia works. — Arthur Rubin | (talk) 19:14, 13 March 2006 (UTC)
- I also notice that "you" (the author of the website referred to) are proud of your links from dmoz.org and Google directory. You should note that the Google directory is a copy of dmoz.org, and the Open Directory Project standards do not require notability or accuracy. — Arthur Rubin | (talk) 19:54, 13 March 2006 (UTC)
Merge with Twin prime?
A valid proof of the twin prime conjecture or just another conjecture?
Euclid’s proof of the potential infinitude of the prime numbers declares that, for the first n prime numbers p1, p2, p3, …, and pn, there is at least one prime number among the natural numbers inclusive between pn + 1 and p1p2p3…pn + 1 since the latter is either prime to all the pi (1 &le i &le n) or is divisible by a prime number greater than pn.
It is clear that the argument applies just as well to p1p2p3…pn – 1. Thus, one could readily assert that there exists “infinitely many” natural number n such that, for the first n prime numbers, both p1p2p3…pn –1 and p1p2p3…pn + 1 are prime numbers. For examples, for n = 2: (2)(3) – 1 = 5 and (2)(3) + 1 = 7; for n = 3: (2)(3)(5) – 1 = 29 and (2)(3)(5) + 1 = 31; for n = 5: (2)(3)(5)(7)(11) – 1 = 2309 and (2)(3)(5)(7)(11) + 1 = 2311; etc. [in general, there are “infinitely many” prime numbers of the form: a power of 2 times a product of powers of odd prime numbers plus or minus 1 &mdash for examples, (22)(3) ± 1 = 11 or 13; (2)(33)(5) ± 1 = 269 or 271; etc.].
The “arguable issue” is whether this contention is already a valid proof of the twin prime conjecture or just another “conjecture” considering that it could not also be actually completely verified nor disproven by a counterexample --- that is, one cannot establish that there exists some natural number y such that, for all natural number z > y, both p1p2p3…pz – 1 and p1p2p3…pz + 1 could not be prime numbers.
Please read my discussion text Prime number: About the incomplete totality of the set of all prime numbers in the Wikipedia article Prime number. [BenCawaling@Yahoo.com] BenCawaling 10:34, 16 April 2006 (UTC)
- This is not a proof. It's plausible -- but I don't think it's even a sketch of a proof. — Arthur Rubin | (talk) 13:58, 16 April 2006 (UTC)
- That is my general idea --- any argument purported to be a "proof" of the twin prime conjecture would just be another conjecture. BenCawaling 07:53, 18 April 2006 (UTC)
- I don't generally like to be blunt, but there are exceptions. That last statement is abject nonsense: if a proof was found it would be a theorem, not a conjecture, just like any other theorem in number theory. In addition, your third paragraph shows that you are very confused. You would benefit from realizing this, and go and refresh your memory as to what a proof is. Meanwhile, such posts are not appropriate for the discussion page of an article on mathematics - this is an encyclopedia which presents accepted results, not alt.philosophy. Elroch 09:31, 18 April 2006 (UTC)
"In 1940, Paul Erdős showed that there is a constant c < 1 and infinitely many primes p such that p′ - p < c ln p where p′ denotes the next prime after p."
All primes are integers. Therepore the difference of two consecutive primes could never be less than one, as this statement implies. Is there a type somewhere in there, or is this just plain untrue? -- He Who Is[ Talk ] 23:16, 26 July 2006 (UTC)
- I got caught with this one already! It is not just p' - p < c, look carefully and you will see it is p' - p < c ln p. I skipped over the natural logarithm of p every time I read it until someone pointed it out to me! Gondooley 23:42, 26 July 2006 (UTC)
- I have changed the layout to make it more clear. (I hope! Let me know what you think!) Gondooley 23:46, 26 July 2006 (UTC)
- No problem. It is humbling isn't it! :-) Gondooley 00:23, 28 July 2006 (UTC)
If this proof were false, then there would be a finite number of primes such that p - p' = 2, and goldbach's conjecture would be rendered false. If this theorem were prooved correct it would be a significant step to prooving Goldbach's conjecture.
The converse is also true. If Goldbach's major or minor conjecture were proved correct, this conjecture would automatically be prooved alongside it.
I would not be surprised if Goldbach's conjecture actually derives from this one, or was produced while attempting to solve it.
Corrected twin primes constant
I corrected 0.66016118158468695739278121100145 to 0.6601618158468695739278121100145. (Am I geeky, or what?) One of the digits had somehow gotten doubled. I also corrected it on the French Wikipedia, but will leave other languages for those more familiar with them than I am. The incorrect number has propagated to numerous non-wiki websites. Keith Lynch 21:51, 15 October 2006 (UTC)
Is the twin prime conjecture important? Does it relate to other results in number theory, in math, in computer science, in theoretical physics, ...? All that I see here is a mention of sieve theory --- would it be a stretch to relate the TPC to integer factoring/discrete logarithm algorithms, and hence to encryption?
Since this problem is so famous and so accessible to the non-mathematician, an encyclopedia needs some explanation of why mathematicians study it. If the only reason is "it's old and simple but surprisingly hard", then we'll have to say that. Joshua R. Davis 15:53, 5 February 2007 (UTC)
- I don't know a relation to integer factoring, discrete logarithm algorithms, or encryption. An explanation of why it is studied must be based on reliable sources. Maybe it's partly because "it's old and simple but surprisingly hard", but unsourced speculation from editors would be original research. I don't think an encyclopedia "needs" to explain it, although it might be nice. The twin prime conjecture by itself doesn't seem to me to have great significance for other things, but maybe techniques used in a proof could be used for many other things - and maybe the first proof (if one is found) will just be a special case of a much more general theorem. PrimeHunter 16:16, 5 February 2007 (UTC)
- I agree with the need for sources, and I don't intend to invent applications for the TPC. That's why I was asking for them here. But Wikipedia:Make technical articles accessible and Wikipedia:Manual of Style (mathematics) (see also Wikipedia:WikiProject Mathematics/Motivation for discussion) encourage motivation, "practical applications", etc. Cheers. Joshua R. Davis 16:46, 5 February 2007 (UTC)
I moved the following fragment which was added to the article by User:Karl-H here.
- Using Wiener-Ikehara theorem, the 'Hardy-Littlewood' conjecture is equivalent to the asymptotic behavior:
Or equivalently the series:
has a pole at s=1 with residue
You can take a look at the paper ' Distributional Wiener-Ikehara theorem and twin primes' by Jacob Korevaar, published ( i downloaded the e-print but it's published) at Indag. Mathematica N.S 16 (1) 37-49 in case you have a doubt i can send you the .pdf --Karl-H 15:01, 24 March 2007 (UTC)
- Then you should have referenced Korevaar, explained that Λ is the von Mangoldt function, and made an effort to embed this remark in the overall narrative. Moreover, it seems to me that the paper just states that the conjecture is equivalent to
- That's apparently supposed to be obvious. The Wiener-Ikehara theorem comes in later. I do hope that anything you add to the article is correct. -- Jitse Niesen (talk) 13:18, 29 March 2007 (UTC)
There exists a twin prime pair between n and 2n, for all integers n > 6.
In June 2007, I made this conjecture after a long search, first by hand, and then by computer, revealed no counterexamples to my conjecture.
PhiEaglesfan712 - June 26, 2007
- Talk pages are for discussing improvements to the article and not for general discussion of the article topic. If no reliable source has mentioned your conjecture then it's unsuitable for the article. That being said, twin primes appear to be fairly common and your conjecture seems likely to be true. I just verified it for n < 10100 in a few minutes with PARI/GP. PrimeHunter 22:09, 26 June 2007 (UTC)
- Nice conjecture. I'm not aware of any prior paper. I suggest you publish it. Giftlite 22:31, 26 June 2007 (UTC)
- It's easy for amateur mathematicians to make plausible prime conjectures which nobody can currently prove or disprove. There are probably thousands of such conjectures on the Internet (I have seen hundreds). I think it would be very hard to get a serious math journal to publish this conjecture. It's also quite possible that it follows from other published conjectures about twin prime occurrences. PrimeHunter 23:23, 26 June 2007 (UTC)
Your conjecture is well-known. It follows, for instance, from the Hardy-Littlewood conjectures with a reasonable error term by an argument identical to how one shows that Bertrand's Postulate follows from the Prime Number Theorem (with an error term). -MM 10/3/08 —Preceding unsigned comment added by 18.104.22.168 (talk) 15:47, 3 October 2008 (UTC)
Claim to be from Euclid
I have removed the claim that this conjecture was proposed by Euclid. This statement was added a long time ago by 22.214.171.124. The sole other contribution made by this user is an edit to Euclid that is evidently nonsense. Eric119 (talk) 04:50, 8 April 2008 (UTC)
- I agree with the removal. In  Daniel Goldston (or somebody identifying as him) says: "No one really knows if Euclid made the twin prime conjecture. He does have a proof that there are infinitely many primes, and he or other Greeks could easily have thought of this problem, but the first published statement seems to be due to de Polignac in 1849". PrimeHunter (talk) 12:59, 8 April 2008 (UTC)