Talk:Goldbach's conjecture

Historical claims

I believe the statement given is that in the letter from Euler to Goldbach, the letter from Goldbach to Euler said "Es scheinet wenigstens, daß eine jede Zahl, die größer ist als 2, ein aggregatum trium numerorum primorum sey."

(A scan of the letter is at, [[1]])

This seems to be Goldbach's weak conjecture. Are you saying that the strong conjecture discussed in this article was not made by Goldbach, but by Euler? AxelBoldt 21:26 Nov 21, 2002 (UTC)

Yes. My German's not too strong but as I understand it, it isn't the weak conjecture as it says every integer (not just odds) can be expressed as the sum of three primes. My understanding is that in his reply Euler simplified it into the form we use now, although I can't remember where I read that. (possibly Hardy and Wright's intro to number theory). --Imran 00:41 Nov 22, 2002 (UTC)

I guess the quasi Goldbach Conjecture was proved by Alfred Rényi in his PhD thesis in 1947 when he worked with Vinogradov in Leningrad. Vamos 20:50 Oct. 11, 2003 (UTC)

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The conjecture had been known to Descartes.

Without further information (not even a year) this statement is useless. What were Descartes' results? Why isn't it called 'Descartes' conjecture'? Any references?
—Herbee 02:17, 2004 Mar 6 (UTC)

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Waring

From [2]:

Waring produced a number of theorems and problems, some without proofs, just statements noting that he knew what was true. For example he published Goldbach's Theorem (every even number is the sum of two primes and every odd number is either prime or the sum of three primes) before Goldbach. He is best remembered for Waring's Problem (each positive integer is the sum of four squares, nine cubes, 19 fourth powers, and so on), later generalized and proved by David Hilbert in 1909.[Weeks 1991]

—DIV (128.250.80.15 (talk) 00:14, 2 May 2008 (UTC))

Trend

"Since this quantity goes to infinity as n increases" is inaccurate. That is not specifically why Goldbach's is probably true. For instance, I could have a function that's 1000 at a billion, 1005 at a trillion, 1010 at a quadrillion, etc. This would go to infinity as n increases, but if it represented the expected number of ways to be able to do something, it would be almost irrevocable that there was a number somewhere that just managed to miss all its chances. No, it's actually got to have to do with how fast this quantity increases, in which case we should calculate the chances that it's zero for a given number and then sum to infinity. dbriggs 24.218.212.123 18:23, 25 February 2007 (UTC)

Well, I've done a rough calculation, it's not hard, and it's pretty easy to see that the total chance of a miss goes down exponentially with n. If Goldbach's conjecture doesn't fail in the first hundred, it is very unlikely ever to fail.
Perhaps this is not the right forum, but I'm curious: Mathematicians - shouldn't this probability argument make it seem likely that Goldbach's conjecture has no proof at all? A finite proof should, after all, reveal some deep mechanism in the infinite number system that causes a fact to be true. If this particular fact is true, quite likely, simply as a coincidence - because there "aren't enough numbers" - probably there is no deeper reason.
That seems like a neat idea to me; is it true, and is it well known? —Preceding unsigned comment added by 162.129.251.22 (talk) 18:47, 26 February 2009 (UTC)

Correct me if I'm wrong, but Goldbach's conjecture can't be undecidable. If it were, then we could never tell if it were true or false. Thus no counterexample exists, as if it did we could (theoretically) find it, which would constitute a disproof of the conjecture. But if no counterexample exists, then the conjecture is true. Only conjectures that can't be disproved by counterexample (e.g. there are only a finite, but unspecified, number of such and such) can be undecidable. — Preceding unsigned comment added by 131.111.185.74 (talk) 22:26, 24 October 2011 (UTC)

Prenex normal form

Is the In prenex normal form correct? olivier 11:13 Feb 14, 2003 (UTC)

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Moved the formula here:

In prenex normal form:

∀ n ∃ p ∃ q ∀ a,b,c,d [(n>2,a,b,c,d>1) ⇒ ((p+q=2n) Λ (ab ≠ p) Λ (cd ≠ q))]

• I don't see the point of this formula; the statement is perfectly clear without it and only a computer would be helped by this formalization. If anything, it could be added as a (defective, see below) example on the prenex normal form page.
• Not even a computer would be helped by the formula, since it is not well-formed formula. Commas are not allowed, especially if used in different senses.
• The unicode characters are not the correct ones and are not visible on Internet Explorer 6.0. AxelBoldt 01:00 Feb 22, 2003 (UTC)

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Thinking about it, you are right. reading carefully, the formula is wrong. Or perhaps not wrong, just not as sharp as it might be. TeunSpaans 22:00 Feb 22, 2003 (UTC)

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Meaning of 'sufficently large'

In 1966, Chen Jing-run showed that every sufficiently large even number can be written as the sum of prime and a number with at most two prime factors.

"What does 'sufficently large' mean?" is a likely question for a reader of this article. Wouldn't it be better saying that there is some number n such that all numbers greater n fullfil Goldbach's conjecture, and adding that nobody knows how big the number n is. -- mkrohn 15:48 Apr 22, 2003 (UTC)

Marco,

Chens result is not identical to Goldbachs conjecture, not even for every number> some unknown number n. -- Anonymous

Mkrohn accurately formulates what mathematicians mean by the phrase "sufficiently large". I'll add a link to sufficiently large to make this clear for everyone.

—Herbee 03:43, 2004 Mar 6 (UTC)

Lawson's Conjecture

I removed the following text: Lawson's Conjecture- for every positive integer (I) greater than 2 there exists a pair of prime numbers a symmetric distance from I. That is, for every I greater than 2 there exists an integer n such that (I+n) and (I-n) are primes. bill_lawson@carleton.ca The conjecture is trivially disproved in the case I=3: the only prime smaller than 3 is 2, which implies that if n exists it must be 1, but n cannot be 1 since 4 is composite. - GaryW 20:24 May 1, 2003 (UTC)

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I have confirmed the alleged "Lawson's conjecture" to up to 100,000 (with a C++ program; Note: NOT 100% VERIFIED yet), and I am pretty certain this has been conjectured (or proven?) before. Any qualified mathematician/number theorist here to shed some light over the matter?

Also, the statement "A proof of this conjecture would prove Goldbach's conjecture." looks suspicious at first glance? anyone? Rotem Dan 10:24 May 4, 2003 (UTC)

Found it -- Euler primes :), this is not relevant. and seems like an old conjecture.. Rotem Dan 10:46 May 4, 2003 (UTC)

I have moved this into Lawson's conjecture.

Yao Ziyuan's Conjecture

After a simple search within 100,000, Yao Ziyuan from Fudan University found that only 4, 6, 8, 12 (even number) can be represented as the sum of one and only one pair of primes. So he made another conjecture on Apr 16, 2004: Every even number greater than 12 can have more than one representation of different pairs of primes. LOL.

This practice demonstrated that we can easily make as seemingly beautiful conjectures as the Goldbach one, as many as possible. This lowers the uniqueness of the Goldbach conjecture and makes it much less significant to prove merely one such conjecture, even if it is eventually proven.

The above was removed from the article as it's not about Goldbach's conjecture. Either this deserves its own article or its not wiki-worthy. DJ Clayworth 13:55, 16 Apr 2004 (UTC)

Um ... that's a bit of a strong statement. One of the best reasons to believe Goldbach is that the number of representations should grow like N/(log N)² (and so should tend to infinity). The YZ conjecture is a minor straw-in-the-wind piece of the puzzle, therefore. That is, with any explicit error term in the number of the representations, one could predict just this, for N >> 0.

Charles Matthews 20:16, 16 Apr 2004 (UTC)

I didn't analyse the maths underneath this. It looked as though it's presence was meant to say "see, there are lots of conjectures"!. Feel free to put it back if you disagree. DJ Clayworth 20:25, 16 Apr 2004 (UTC)

Another Conjecture (Fall out from Yao Ziyuan's Conjecture)

After a simple search, by hand, within 10,000, I found that only 4 (2 + 2) and 6 (3 + 3) (even number) cannot be represented as the sum of two distinct prime numbers. Also, I found that only 8 (3 + 5), 10 (3 + 7), 12 (5 + 7), 14 (3 + 11), and 38 (7 + 31) can be represented as the sum of two distinct primes in one and only one way. So, I conjecture on July 13, 2007 that:

• 1) Every even number greater than 6 can be represented as the sum of two distinct prime numbers.
• 2) Every even number greater than 38 can be represented as the sum of two distinct primes in more than one way. (In other words, I am conjecturing that 38 is the largest number that can be represented as the sum of distinct primes in only one unique way.) PhiEaglesfan712 18:36, 13 July 2007 (UTC)

• • See also http://www.research.att.com/~njas/sequences/A000954 and http://www.research.att.com/~njas/sequences/A056636. They don't require distinct primes. PrimeHunter 18:57, 13 July 2007 (UTC)

Goldbach equivelent to Lawson

The following argument should show that Lawson's conjecture is equivalent to Goldbach. Assume Goldbach, and let n be the given integer in Lawson. Then 2n is even, and there exists two primes p and q such that 2n=p+q. Assume p is less than or equal to q, and take l=(q-p)/2. Noting that n=(p+q)/2, observe that n-l=p, and n+l=q. Assume Lawson, and let 2n be the given even number in Goldbach. Then there is an l such that n-l=p and n+l=q are primes. Clearly, 2n=p+q. Also note that l need not be non zero, hence garyW's objection. If 2n=p+q and p is even, note that that requires q to be even, and 2n to be 4. Goldbach might be restated as, every even number greater than 4 is the sum of two odd primes, and Lawson might be given as every n larger than 3 has an l such that n-l and n+l are odd primes.

"Later mathematicians" paragraph

I have made significant revisions to the paragraph that begins with "Later mathematicians" and discusses two generalized proofs that would each prove the Goldbach conjecture as a special case. First, the English was rather poor. Also, I'm fairly certain that the original text was actually in error, as the second approach didn't specific any sum of numbers. (It merely said "can be written as [a number] and [a number]".) I think I revised it to match the original intent, but since I am not a professional mathematician, I'd appreciate it if someone more qualified could verify that my revisions state the approaches correctly. -- Jeffq 03:58, 9 May 2004

May I add three remarks:

(A) The two statements: "any odd number not smaller than 7 is a sum of at the most three primes" and "any even number not smaller than 4 is a sum of at the most two primes" are one implying the other, because it is always possible to express "any odd number not smaller than 7 as a sum of 3 and one even number (not smaller than 4)." If one statement is true, so is the other.

(B) The statement 2N = P₁+P₂, where N is an integer not smaller than 2 and P₁ and P₂ are primes, is not contradicting the theory that the closed interval [N, 2N] must contain at least one prime, because the larger of P₁ and P₂ must be in [N, 2N]. Furthermore, since there is likely more than one pair of P₁ and P₂ when N is not smaller than 7, the interval [N, 2N] will likely have not one but two or more primes.

(C) If the expression 2N = P₁+P₂ is re-arranged as:

 N = (P1+P2)/2

 P2-N = N-P1, with P2 not smaller than P1,

one can see that N is a point of reflection about which P₁ and P₂ are each other's mirror image. Suppose P₁, P₂, P₃, ...., P_max are all the primes smaller than N. We can choose N = P_max+1. Then, at least one of Q_k = 2P_max+2–P_k will be a prime, if Goldbach's conjecture is true. The choices of P_k are definite and finite, albeit very large, and a prime in the interval [N, 2N] is assured. 64.231.5.139 23:23, 15 September 2006 (UTC)

Yes, yes. That is very true. ∀∃"e_i"∴±{o_1,o_2}⇒{p_1,p_2}∵∀∃o_1+o_2=e_x∵∀e=o_i*2-71.159.34.238 03:59, 13 December 2006 (UTC)

Claims of proofs

What to do about these?

(A) I would say the Pogorzelski claim from 1977 has nothing encyclopedic about it.

(B) The claim from Belarus - any support at all for this rumour?

(C) The claim on behalf of a student. The Andrew Wiles quote surprises me; this is not the usual way of doing business at Annals of Mathematics.

In fact all of these could be taken out, without some better support.

Charles Matthews 15:32, 16 Sep 2004 (UTC)

I agree these are all rather dubious claims, but am unsure what to do about them; for now, I've moved them into their own section. Terry 06:27, 28 Sep 2004 (UTC)

It's not getting any better, and failed attempts at proof have no encyclopedic value. I've moved them all here (follows). Charles Matthews 20:13, 17 Dec 2004 (UTC)

For instance:

1. H.A. Pogorzelski circulated a proof of the Goldbach conjecture in 1977, but this work is not generally accepted in mathematical circles.
2. Viktar Karpau (Victor Karpov), a mathematician from Belarus, allegedly found a proof of Goldbach's conjecture which was published in September 2004.
3. A student at the University Of London has claimed that he has found proof of the Conjecture. Andrew Wiles, an editor of Annals of Mathematics who proved Fermat's Last Theorem in 1994, has seen part of the proof and has said it looks very promising.
4. A simple 8-page proof of the Goldbach Conjecture, discovered in early October 2004, by Jay Dillon, has been claimed and will be submitted to a journal when typesetting is completed, expected by early January 2005. (Dillon recently published a simple geometric proof of Fermat's Last Theorem in WSEAS Transactions on Mathematics, July 2004; a condensed one-page proof of FLT using the same geometric method is also accepted by WSEAS, designated WSEAS paper no. 10-352, not yet published.) An even briefer proof of the Goldbach Conjecture, and a similar brief proof of the Odd Goldbach Conjecture, have been prepared but not yet verified.

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Hi guys. In the origins section look at these lines:

So today, Goldbach's original conjecture would be written: Every integer greater than 5 can be written as the sum of three primes.

Should that 5 not be a 2? Also should "three primes" in that sentence not be "two primes"? As we are talking "today" when one is not a prime? I dont know but those look like typos to me.

Also I would appreciate it if anyone can help flesh out the 1 no longer prime page (look in the history) which I thought I'd create to help explain why Goldbach considered 1 to be a prime. (And who else possibly) Key things to add are the date this became formal in world mathematics and the exact reasoning since I am only going to give examples of pattern breaking etc. It will be a dodgy inductive article until someone more familiar comes in.

Ta Cyclotronwiki 27 April 01:33 Taipei

There are many additional - see this website to search for recorded documents via US Library of Congress US Libr Congress search--Billymac00 (talk) 19:34, 16 December 2007 (UTC)

Chinese names

...with the currently best known result due to Chen and Wang in 1989...

I think this is a very bad and somewhat ignorant practice to refer to Chinese people with their surnames only, in the same way as referring to non-Chinese. Most Chinese use only a few most common surnames, so a surname is used by lots of people. And both Chen and Wang are the most common surnames, with millions of people sharing them. It is impossible to find out who these people are with only their surnames. So I ask whoever citing Chinese people to always give their full names just as it is done in Chinese literature. --Small potato 06:22, 24 Jun 2005 (UTC)

Graphs

I added two graphs showing the number of ways in which n can be written as the sum of two primes; one going up to 1000, and the other one going up to a million (showing a remarkable distribution of the function's values). Golbach's Conjecture, of course, is that these functions have no n such that g(n) is zero.

Perhaps the graphs could need some elucidation in the main text. reddish 16:16, 22 March 2006 (UTC)

At last a proof?

Can someone look at this: http://arxiv.org/abs/math/0701188 and confirm or refute? —The preceding unsigned comment was added by 132.66.222.96 (talk) 21:56, 8 April 2007 (UTC).

Better give this link http://arxiv.org/abs/math.GM/0701188 I haven't read the paper and i think the authors have to start by deleting the first page with the graph etc. and start the paper from the: "One can conclude by induction that...". The "observation" that every even is the sum of two odd numbers is low-level. -- Magioladitis 01:08, 28 May 2007 (UTC)
I don't get the "low-level" comment, I think such a property is fairly integral to a proof.--Billymac00 00:18, 17 September 2007 (UTC)

Notice the formula (1) in page 2 is wrong. Take for example n = 3. Since 6 = 1+5 = 3+3 the formula should give 2 but it gives 1. They' ve forgot a +1 at the end i.e. the formula should be (n + n mod 2) /2, (without the -2 on the nominator). But don't worry, the formula doesn't affect the rest of the paper. The figures are really unnecessary if the paper is intended to be published in a scientific magazine. The property you are referring it's, let's say, "obvious" -- Magioladitis 00:08, 9 October 2007 (UTC)

Check here. There are many proposed proofs. -- Magioladitis 00:11, 9 October 2007 (UTC)

Heuristic justification

I've reformatted the paragraph which opens this section to divest it of weasel words. I however remain unsure that it's a necessary inclusion. Digby Tantrum 17:55, 21 May 2007 (UTC)

I think it's excellent, as redone. Very understandable by the casual reader, which is the role of an introductory paragraph, without being misleadingly simplistic. - DavidWBrooks 21:37, 21 May 2007 (UTC)

Well, thank you. But it should be stressed that this retains much of the original version of that paragraph, which I didn't author. Mark H Wilkinson 10:14, 22 May 2007 (UTC)

Here's an even stronger version of the Goldbach conjecture

All even numbers greater than 6 can be written as the sum of two distinct prime numbers.

I believe the following is true : which looks pretty much like that above

All Composite numbers greater than 6 can be written as the sum of distinct prime numbers. —Preceding unsigned comment added by 85.80.164.222 (talk) 23:26, 8 September 2007 (UTC)

Actually, that's not equivalent. If you meant to say two distinct prime numbers, that's easily disproven; if n is an odd composite number and n-2 is also composite, then n cannot be written as the sum of two primes. 27 and 35 are the first two examples of this. If you meant any number of distinct prime numbers, I would conjecture that's true of all sufficiently large numbers (not just composite numbers); 11 may be the largest number for which it's not true. --Mwalimu59 01:25, 9 September 2007 (UTC)

I guess if the extended Goldbach Conjecture is true, then the statement "All numbers greater than 6 and not 11 can be written as the sum of distinct prime numbers" is also true. Check out Kerry M. Evans proof of Goldbach Conjecture. http://geocities.com/carryme47714/ —Preceding unsigned comment added by 87.52.72.173 (talk) 14:26, 10 September 2007 (UTC)

Checked it out and found an error. It says "if m is a composite number, then and only then (m-1)! = 0 (mod m)". This is not true - counterexample is m=4. Gandalf61 14:45, 10 September 2007 (UTC)

Ah yes. Gandalf, absolutely right. if you have time, could you give me two or three more counterexamples? I want to look up the sequence in the OEIS. Anton Mravcek 16:31, 17 September 2007 (UTC)

The only counter examples are 1 and 4. This observation for m>5 is mentioned in Factorial#Number theory. PrimeHunter 19:41, 17 September 2007 (UTC)

Oh, so much for that. Oh well, it does point out an interesting tidbit about 4, the "primest" composite number. Anton Mravcek 00:13, 18 September 2007 (UTC)

Equivalent

"On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) [1] in which he proposed the following conjecture:

   Every integer greater than 2 can be written as the sum of three primes.

He considered 1 to be a prime number, a convention subsequently abandoned. A modern version of Goldbach's original conjecture is:

   Every integer greater than 5 can be written as the sum of three primes.

Euler, becoming interested in the problem, answered by noting that this conjecture would follow from a STRONGER version,

   Every even integer greater than 2 can be written as the sum of two primes,

adding that he regarded this a fully certain theorem ("ein ganz gewisses Theorema"), in spite of his being unable to prove it." - Wikipedia

This seems to me to be a mistake. If, as Goldbach stated, every integer greater than 5 is the sum of three primes, then consider consider even integers greater than or equal to 6. They must be the sum of three primes, hence one of the primes must be equal to two, hence the previous even number is the sum of two primes. Conversely as Euler remarked, if 4,6,8, etc. are sums of two primes then 6 and 7, 8 and 9, 10 and 11 are the sum of three, etc. —Preceding unsigned comment added by 82.32.73.115 (talk) 23:30, 8 January 2008 (UTC)

importance?

The article might give some indication as to whether the conjecture is known to be of any importance. E.g., do any other results depend on it? IIRC I've heard the Goldbach conjecture referred to as an example of how there are undoubtedly many mathematical claims that can be easily stated, but that are of no intrinsic interest and will never be proved.--76.93.42.50 (talk) 21:34, 18 March 2008 (UTC)

I do not think any other results of some importance depend on it. However, the conjecture may be important for its intrinsic mathematical interest (subjective and very hard to assess, but nevertheless real) and because of how it is connected to other interesting mathematical research. It is somewhat clear that knowing with certainty whether the conjecture holds has no practical significance; it will not help to test for primality like a proven GRH would. The problem is that any statement of this nature will reflect a point of view and needs to be attributed to a mathematician of repute.  --Lambiam 09:53, 19 March 2008 (UTC)

Known to be true by verification through a big number

IANAM (I am not a mathematician). I wouldn't think to look in the "Rigorous results" section for the results of a brute force attack on the conjecture.

Well before I got to the "Rigorous results" section, I wondered: "How far has anyone gone to check this by 'experiment'?"

I eventually found that:

• The conjecture is known to be true through what seems like a very large number to me.
• That number is big enough for my purposes.

A separate section for the result of the latest brute force attack would be helpful to a layman. A clickable title in the list of contents would help them cut to the chase. Preferably before the 'Heuristic justification' section. Telling the reader that the conjecture has been verified extensively builds credibility. It may pique their interest for the details in the 'Heuristic justification' section.-Ac44ck (talk) 15:58, 25 April 2008 (UTC)

The Use of the Conjecture?

Is there any practical utility in proving the conjecture (other than a sense of accomplishment?) are there any famous results that depend on the conjecture and can one use this for cryptography or something of the sort?Philosophy.dude (talk) 09:37, 7 August 2008 (UTC)

Speaking as a non-mathematician, I believe any effect would depend on the proof - if it was based on a previously unsuspected connection that could have ripple effects throughout math, then it's great. If it's just a slog like the computer-based proof of the four-color theorem, then no. - DavidWBrooks (talk) 13:52, 7 August 2008 (UTC)

I'd be very surprised if any proof at all existed; I think it's true, but for accidental, probabilistic reasons. So if anyone did come up with a deeper reason it was true, it would probably be quite interesting. A counterexample, on the other hand, would not be very interesting. Tualha (Talk) 14:47, 3 September 2008 (UTC)

A proof would touch on the density of the primes because a counterexample would show some region without any near enough primes. If I were to try a proof, I'd assume a counterexample N, chosen to be the smallest number for which the conjecture does not hold then work up the size of the prime factorization. (Cannot be a single prime factor because N is a large even number. Cannot be two prime factors because then N = 2 * P which means that N = P + P, and so on.) Hcobb (talk) 21:06, 23 May 2011 (UTC)

@Tualha A counterexample might not be very interesting to you, but it would amaze me. So many mathematicians have worked on this over the years and have found so many examples, yet the Conjecture turns out to be false?! Could anyone fail to find such an event interesting? David Spector (talk) 01:23, 14 July 2011 (UTC)

People said that Fermat's last theorem was unprovable, and a useless result, and were wrong on both counts. (Useful result because really, Andrew Wiles didn't prove FLT per se, he proved some obscure conjecture by two Japanese mathematicians whose names elude me about modular forms and elliptical curves, on which a lot of other work in that field depended. Someone else, who somewhat sadly doesn't get any of the credit, had first proved FLT assuming the other conjecture was true.) Actually, if there is a counterexample it would probably say something deep about the distribution of primes, which is in turn related to Riemann hypothesis etc etc. The heuristic argument, which assumes primes are distributed randomly, suggests that it is vanishingly unlikely that a counterexample larger than, say, a million exists, so if there were a counterexample then it would almost certainly be the case that the distribution of primes wasn't as random as we thought. — Preceding unsigned comment added by 92.27.55.215 (talk) 19:18, 26 April 2012 (UTC)

Original claim was all integers

By my and Babelfish's reading of Goldbach's original German (see note 1 above: "Es scheinet wenigstens, daß eine jede Zahl, die größer ist als 2, ein aggregatum trium numerorum primorum sey"), Goldbach's original claim was for all integers, not just odd integers. So I've deleted this sentence "Note Euler's formulation implies Goldbach's original version, but not vice-versa" this reference which incorrectly claims it was for only odd integers.[3] p.s. This was the (mis-)understanding behind my edit yesterday. Peter Ballard (talk) 10:26, 14 August 2008 (UTC)

Counterexample

I have discovered a truly remarkable counterexample, which this editor window is too small to contain. Tualha (Talk) 01:30, 2 September 2008 (UTC)

Ha ha. PrimeHunter (talk) 02:29, 2 September 2008 (UTC)

"Weak" and "strong" forms

The article ("Origins" section) initially says that these two forms are "equivalent", but then goes on to imply that they are not equivalent -- the strong form implying the weak form, but not necessarily vice versa? Is it just me, or does this section not really make sense? Matt 04:00, 12 October 2008 (UTC) —Preceding unsigned comment added by 86.136.27.87 (talk)

It's not just you; I just noticed that contradiction myself. LaQuilla (talk) 18:55, 1 January 2009 (UTC)

Elementary proof of a very weak form of Goldbach's conjecture

I am going to remove this reference. In this reference, the following is proved. Given n distinct natural numbers, there are at least 2n-1 different 2-term sums from these numbers. This is true, the proof is correct, the result is well known. But it has nothing to do with Goldbach's conjecture. Kope (talk) 16:14, 31 December 2008 (UTC)

Polignac's Conjecture

From the proof given, $S=p_{1}+p_{2}=2p_{2}+d$, where $d=p_{1}-p_{2}$ For $S$ to have any possible even value, $d$ must have any possible even value. This requirement is the exact claim of Polignac's conjecture, so Polignac's Conjecture implies Goldbach's Conjecture. —Preceding unsigned comment added by 70.129.186.76 (talk) 01:38, 8 January 2009 (UTC)

This is both original research and a wrong proof - but you don't have to agree it's wrong. Without a reliable source it doesn't belong in the article so I have reverted it. PrimeHunter (talk) 02:47, 8 January 2009 (UTC)

The error is that the "proof" assumes p₂ is fixed, whereas it isn't - it depends on d (even assuming Polignac's conjecture). But even if the proof of equivalence were correct, PrimeHunter's point is that it would still need to have a source. I have removed the "proof" from the article again. Gandalf61 (talk) 15:48, 8 January 2009 (UTC)

While $p_{2}$ isn't fixed, if $d$ can only have specific values, $S$ can only have specific values as well. For example, if $d$ is only 2, the only values of $S$ that will be possible are 6, 8, 12, 16, 24, ..., even ignoring the fact that many of these don't have a possible prime $p_{1}$ that results in $d=2$, which means that the possible valid values for $S$ with $d=2$ leave even more gaps. By extending this argument, if we let $d$ be contained in any finite set of positive even integers, there will be some even numbers that cannot be created. Therefore, $d$ must be able to have any positive even value for Goldbach's conjecture to be true. Since this statement is what is conjectured by Polignac's conjecture, Polignac's conjecture implies Goldbach's conjecture.

As for your use of WP:RS:

WP:IAR

Since this proof is correct and would add new and important information to the article, and since each step can be verified logically and mathematically, it can be added regardless of WP:RS. --Oboeboy (talk) 16:07, 8 January 2009 (UTC)

Let me try to explain the error another way. Polignac's conjecture says that for each even d there are an infinite number of pairs of consecutive primes {p₁(n,d), p₂(n,d)} such that p₂(n,d) - p₁(n,d) = d. For each such pair of primes we can create the even integer p₁(n,d)+p₂(n,d). But there is no guarantee that the set {p₁(n,d)+p₂(n,d)} covers all even integers. Maybe from some point onwards p₁(n,d)+p₂(n,d) is always a multiple of 10 (for example). Then Polignac's conjecture would be true but would not imply Goldbach's conjecture.

But, in any case, WP:NOR is such a fundamental Wikipedia policy that you are very shaky ground if you try to use WP:IAR to trump it.

I see you have added the "proof" again. I am sure someone else will revert your change fairly soon. Be careful not to break WP:3RR. Gandalf61 (talk) 16:24, 8 January 2009 (UTC)

WP:3RR#exceptions

Reverting vandalism is allowed. Since you are removing valid information, what you are doing is vandalism, so WP:3RR doesn't apply and I can revert it as much as needed.

Anyway, my proof in fact doesn't violate WP:NOR, since each step uses nothing but one or more mathematical axioms which are easily sourced, and since WP:NOR states:

The "No original research" rule does not forbid routine calculations (e.g. adding or subtracting numbers, rounding them, calculating percentages, converting them into similar units, putting them on a graph, or calculating a person's age) that add no new information to what is already present in the cited sources.

Furthermore, my proof is valid, because if $d$ could only be, for example, 2, then only certain values (6, 8, 12, 16, 24, ...) would be possible. Similar problems occur when trying to use any finite set of primes, even if you consider all values of $2p_{2}+d$ regardless of whether $p_{2}+d$ is also prime. Therefore, no finite set of even values of $d$ will be able to produce every even number, so an infinite set of possible values for $d$ is necessary to be able to produce every even number. --Oboeboy (talk) 16:40, 8 January 2009 (UTC)

Yes, an infinite set of possible values for $d$ may be necessary to produce every even number, but you have not shown that it is sufficient. But good luck with filling in that missing step. Gandalf61 (talk) 16:50, 8 January 2009 (UTC)

I concur the proof given by Oboeboy is mathematically incorrect, and including it on the article is both an embarrassment for wikipedia and a violation of wikipedia policy. The proof is not even using Polignac's conjecture (that each prime gap occurs infinitely often), but merely that for each "gap" d, it occurs at least once and not even as a prime gap, just as a difference of primes. Such a weak conjecture obviously says nothing about writing small even numbers as sums of primes. Some of the "proofs" given seemed to try and show that Goldbach implies Polignac, but they only attempt to prove the much weaker conjecture I mention, that infinitely many even numbers are a difference of primes. JackSchmidt (talk) 18:47, 8 January 2009 (UTC)

weak goldbach -> at most 4 primes??

"In fact, resolving the weak Goldbach conjecture will also directly imply that every even number n ≥ 4 is the sum of at most four primes."

shouldn't weak goldbach imply every even n>=4 is the sum of at most THREE primes? Rainjacket (talk) 23:18, 24 February 2009 (UTC)

The weak conjecture is all odd numbers greater than 7 are the sum of three odd primes, so to that result you can add 3 (a fourth prime) each time and it would give you every even number from 10+. --86.139.65.151 (talk) 17:58, 21 April 2009 (UTC)

But what if 3 is already one of the primes being used? - DavidWBrooks (talk) 18:21, 21 April 2009 (UTC)

That's okay. There is no requirement that the primes be unique. --mwalimu59 (talk) 19:50, 21 April 2009 (UTC)

D'oh! (slaps forehead) ... - DavidWBrooks (talk) 20:07, 21 April 2009 (UTC)

difference of primes

Arguably this comment belongs on the talk page for Polignac's conjecture but seems justifiable to put it here, and a lot more people are watching this page. Is it an open conjecture (and if so, does it have a name and a history) whether or not every even number is the difference of two primes? (Polignac is much stronger, stating that every even number occurs infinitely often as the difference of consecutive primes) —Preceding unsigned comment added by 99.175.86.5 (talk) 13:14, 13 May 2009 (UTC)

This is still an open question. It would follow as a special case of different well-known conjectures, for example the k-tuple conjecture with k=2, but I have not heard a separate name for it. In 2004 I proved it experimentally for even numbers up to 10¹².[4][5] PrimeHunter (talk) 15:02, 13 May 2009 (UTC)

Estimating # pairs adding to some E

this ref [6] seems to have derived a fairly good way to estimate the pairs.--Billymac00 (talk) 03:43, 30 June 2009 (UTC)

"None is" or "None are"

Sorry, oh Grammar Nazi, but "none" does not automatically take a singular verb; the plural is equally acceptable, depending on whether you consider the word "none" as "not one" (singular) or "the opposite of all" (plural). There are plenty of sources to support this - you can Google "none singular plural" to find them, or check the NY Times style blog here: [7].

In the sentence being debated, I prefer the plural because it sounds better, although it's not necessary more correct. - DavidWBrooks (talk) 21:58, 2 July 2009 (UTC)

Idiotic

The line starting "1.32..." at the bottom of the paragraph entitled "Heuristic Justification" is idiotic and disjointed. —Preceding unsigned comment added by 86.137.170.8 (talk) 11:21, 25 July 2009 (UTC)

It has now been removed. —Preceding unsigned comment added by 81.148.89.37 (talk) 10:18, 8 August 2009 (UTC)

"The weak Goldbach conjecture is fairly close to resolution."

That seems like an unreasonably bold statement to me. How does one know that an unproved theorem is "fairly close" to being proved? Perhaps something like "There has been significant progress on the weak Goldbach conjecture, which is now known to be true for all n greater than..." would be more appropriate. -- Spireguy (talk) 02:02, 7 August 2009 (UTC)

I have re-worded the remark, making it closer to the truth. —Preceding unsigned comment added by 86.177.50.177 (talk) 12:40, 12 August 2009 (UTC)

Now it contains weasel words "thought by some" -- a source is needed here (who thinks it's close to resolution?), or else the statement should be removed.—GraemeMcRae^talk 18:47, 12 August 2009 (UTC)

In Popular Culture - Carl Sagan and minor copy editing issues

In The Demon-Haunted World: Science as a Candle in the Dark, Carl Sagan says that he asks for a proof of Goldbach's conjecture when confronted with someone who claims to be in contact with advanced alien intelligences (and that his request is always ignored in favor of vague moral prescriptions). He says this would be a good question to ask in order to verify the reality of the advanced intelligence to yourself and others, as it's a simple equation to describe. Worthy? 69.157.6.30 (talk) 03:32, 9 August 2009 (UTC)

A fun fact, but not encyclopedic. -- Spireguy (talk) 01:42, 11 August 2009 (UTC)

This is no worse than some of the Goldbach conjecture#In popular culture entries currently in the article. Agree though, that it is non-notable. Anyway, the representative of an advanced alien culture might reply truthfully in answer to the question: "Do you have a proof of the Goldbach Conjecture?" - "No, we do not." The ability to build spaceships capable of interstellar travel would not necessarily lead to proofs of this kind. Incidentally, Carl Sagan recommended asking aliens if they had a proof of the Goldbach Conjecture or Fermat's Last Theorem in the Skeptical Inquirer in 1987, see [8].--♦IanMacM♦ ^{(talk to me)} 15:51, 25 September 2009 (UTC)

I added reference to a suggestion that the conjecture may be a potential unprovable verity. A copy editor might wish to change the English to potentially unprovable verity, but doing so would change the meaning detrimentally. It is not clear that unprovable verities are entities. If they are, then Goldbach's conjecture might be one. If at some point in the future it becomes true that unprovable verities exist, then it would be reasonable for a copy editor to change potential to potentially. Jaredroach (talk) 01:51, 7 February 2011 (UTC)

Euler's quote

Euler's exact wording from 1742 has been added to the article. In line with Euler's thinking, 1 is considered to be prime, so 2 is a Goldbach number. This goes against modern ideas about the primality of 1, but the quote is given so that Euler's statement of the problem is accurate.--♦IanMacM♦ ^{(talk to me)} 19:47, 23 September 2009 (UTC)

True/False

Are there any people out there who believe that there is a counterexample for Goldbach's Conjecture? Or is it general consensus that it is true? I read that Ramanjuan said that he believed that there actually is a counterexample. My 2 Cents' Worth (talk) 12:17, 20 May 2010 (UTC)

Original Research - Scatter Plot Explanation

I understand your need to make sure that the material posted here needs to meet certain standards. I have taken an unorthodox approach to a number of problems and been able to make progress simply because of that approach. The following papers, although original research, are simple enough to follow and should generate interest.

Although I have contributed to the writing of medical research papers previously, I have never attempted a maths paper frankly because I don't have the necessary knowledge to describe them in a way that mathematicians are expecting. My versions are everyman versions but I have had suggestions that I should present two versions, one like these and one for mathematicians.

http://simplicityinstinct.weebly.com/uploads/4/7/3/3/4733019/goldbachsconjectureprimenumberchannels.pdf http://simplicityinstinct.weebly.com/uploads/4/7/3/3/4733019/completeproject.pdf http://simplicityinstinct.weebly.com/uploads/4/7/3/3/4733019/goldbachconjecturebands.pdf

Anyway, you are welcome to look at these papers and pass feedback to info@simplicityinstinct.com —Preceding unsigned comment added by Simplicityinstinct (talk • contribs) 01:14, 4 August 2010 (UTC)

Is Goldbach's conjecture in EXPSPACE?

According to the EXPSPACE article, L. Berman proved in 1980 that "the problem of verifying/falsifying any first-order statement about real numbers that involves only addition and comparison (but no multiplication) is in EXPSPACE." Wouldn't this class of problems include Goldbach's conjecture? This would seem to be a significant result. 129.22.209.113 (talk) 20:44, 16 August 2010 (UTC)

You can't show that a number is prime without multiplication. - DavidWBrooks (talk) 22:51, 16 August 2010 (UTC)

I think you can. You can show that a positive integer is prime by showing it is not divisible by any smaller integer > 1, and you can test divisibility by repeated subtraction. So you can show that 47 is not divisible by 7 by repeatedly subtracting 7; after 6 subtractions you are left with 5, which is greater than 0 and less than 7, so 47 is not divisible by 7. However, by the same argument, multiplication by an integer is just repeated addition, so perhaps I am missing some subtlety here (perhaps the restriction in the original statement only forbids multiplication by non-integers or by irrational numbers, for exmple). Gandalf61 (talk) 13:02, 7 February 2011 (UTC)

Your definition for divisibility did not use a first order language. Try to say - in a first order language: "for every m,n if m is divisible by n then..." Eliko (talk) 17:19, 7 February 2011 (UTC)

Nicola Fragnito

There is no recognized proof of Goldbach's Conjecture. The link here looks like original research.--♦IanMacM♦ ^{(talk to me)} 11:00, 11 April 2011 (UTC)

I agree it doesn't belong here. You removed one of the links to it from the article. I have removed the other. PrimeHunter (talk) 12:21, 11 April 2011 (UTC)

yet another proof?

http://arxiv.org/abs/1110.3465

"The Goldbach conjecture was proved in this paper. The proof was by contradiction based on the fundamental theorem of arithmetic and the theory of Linear Algebra. First, by an assumption, the Goldbach conjecture was converted into a group of linear equations. Then, by investigating solutions to the group of linear equations, reductions to absurdity were derived to prove the assumption false. Hence, the Goldbach conjecture was proved that even numbers greater than 2 can be expressed as the sum of two primes." — Preceding unsigned comment added by 92.233.40.187 (talk) 08:46, 27 October 2011 (UTC)

Interesting. It is unclear whether Cornell University peer reviews submissions to this archive. Whatever, any proof of Goldbach's conjecture would make the author world famous (although not rich, as strangely it is not one of the Millennium Prize Problems).--♦IanMacM♦ ^{(talk to me)} 09:00, 27 October 2011 (UTC)

ArXiv is not peer-reviewed, it is intended as a preprint archive. While there are a group of editors who reject obvious nonsense, if a paper looks even vaguely correct at first glance, it is accepted. Meaning, any crackpot who can make his work look sufficiently convincing to a brief observer can get a paper up there. That a paper exists in arXiv says nothing about its correctness or legitimacy, and the linked proof should be considered invalid until it passes peer-review (if ever). 128.253.232.252 (talk) 22:03, 4 December 2011 (UTC)

Ummmm, is your chart wrong showing Goldbach numbers ... the part where it shows 2(2) = 4 = (2+2) .... why is that there? Or is it wrong? I ask because 2 ISNT a prime number and a Goldbach number is supposed to be an even number CONSISTING OF 2 PRIME NUMBERS ... shouldn't that chart be showing (2)2 = 4 = (3+1) instead? That chart thats supposed to show "Goldbach numbers" infact shows the number "4" made up of 2 EVEN numbers so why is it on a chart that is supposed to show numbers made of 2 PRIME numbers? It should say for "4" that it equals (3+1) instead of (2+2) or am I missing something?

TY 173.238.43.211 (talk) 06:28, 5 May 2012 (UTC)