Talk:Twin prime conjecture

message: 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)
replies: text: 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)

text: 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)

text: "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)

text: 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)

text: 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)

text: 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)

text: No, I did not say that. You keep trying to get me to say that. --Trovatore 14:48, 6 March 2006 (UTC)
thread_title: Removed text

message: Please discuss at Talk:Twin prime#Merge with Twin prime conjecture? --Trovatore 21:18, 13 March 2006 (UTC)
thread_title: Merge with [[Twin prime]]?

message: 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)
replies: text: 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)
replies: text:  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)
replies: text:  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)
thread_title:  A valid proof of the twin prime conjecture or just another conjecture? 

message: "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)
replies: text: 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)

text: 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)

text: No problem.  It is humbling isn't it! :-) Gondooley 00:23, 28 July 2006 (UTC)
thread_title:  Impossible Statement 

message: 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)
thread_title:  Corrected twin primes constant 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Significance? 

message: 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)
replies: text: Nice conjecture.  I'm not aware of any prior paper.  I suggest you publish it.  Giftlite 22:31, 26 June 2007 (UTC)
replies: text: 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)
thread_title:  My Conjecture 

message: I have removed the claim that this conjecture was proposed by Euclid. This statement was added a long time ago by 68.5.186.133. 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)
replies: text: 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)
thread_title: Claim to be from Euclid

message: The article lacks a reasonable History section. Is this the oldest unsolved conjecture in mathematics? Albmont (talk) 13:39, 20 October 2008 (UTC)
thread_title:  History