Talk:Goldbach's conjecture/Archive 1

Archive 1

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)

The theorem that Godel proved was undecidable is clearly true. In the case of Goldbach's conjecture, a proof that it was unprovable from Peano arithmetic would indeed be a proof that it was true, just not one expressible without metamathematics.--Prosfilaes (talk) 01:27, 23 January 2013 (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?

Its so trivial, that I wrote one that ran to 2^32-1, and with access to a good math library ( finally !) for GNU C, I was able to run the program for 2^64-1. 100% Verified, but then its been verified up to 4*10^18, so it is only a excercise in progamming, not anything significant. — Preceding unsigned comment added by 67.180.156.92 (talk) 07:52, 29 December 2012 (UTC)

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)

I wrote and ran a program that tested it to 2^32-1, and it proved true, then with an improved math library, I was able to extended it to 2^64-1, or 18446744073709551615. However, this is an experiment, not a proof. — Preceding unsigned comment added by 67.180.156.92 (talk) 07:59, 29 December 2012 (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)

links just above are dead, refer to http://oeis.org/A000954 and http://oeis.org/A056636--Billymac00 (talk) 22:18, 4 November 2012 (UTC)

"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)

It's not a contradiction. The strong form trivially implies the weak form, and it's now known to be true (but not trivial) that if the weak form is true, the strong form is also true. 2.25.134.93 (talk) 18:54, 31 December 2012 (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)

You are quite right. The weasel words have been removed. — Preceding unsigned comment added by 80.82.65.25 (talk) 14:19, 19 October 2012 (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)

I wonder why this was published in JP J. Algebra Number Theory Appl. The paper's references look very bad. -- Magioladitis (talk) 00:07, 5 November 2012 (UTC)

Apparently you have made a confusion between "JP Journal of ..." and "Journal of ... ". The "JP" at the beginning make a great difference! D.Lazard (talk) 11:06, 5 November 2012 (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)

(removed) thought 2 was'nt a prime number - it is

173.238.43.211 (talk) 06:37, 5 May 2012 (UTC)

Restating the logical version

Seeing as how a = x + b has a unique solution, wouldn't it make more sense to say there exists a p such that p and n - p are both prime? OneWeirdDude (talk) 01:06, 18 May 2012 (UTC)

Number Theorem I

(a+n)=2n-(n-a)=2n+(a-n) ; 1 is prime. Is allways true for all numbers (n+a) in N

Proofs direct Goldbach, Polignac and Levys Conjecture. 1 is here prime

2n = (n+a)+(n-a) Goldbach ; (n+a)-(a-n) Polignac and (a+n)= 2n+(a-n) is Levys Conjecture

=>

2 = (0+1)+(1-0)

4 = (1+2)+(2-1)

6 = (2+3)+(3-2)

8 = (1+4)+(4-1)

10 = (2+5)+(5-2)

12 = (1+6)+(6-1)

14 = (4+7)+(7-4)

16 = (3+8)+(8-3)

18 = (2+9)+(9-2)

20 = (3+10)+(10-3)

22 = (6+11)+(11-6) . . 2n = (a+n)+(n-a)

=> Is ever true while 2n=2n

Every Natural Number >1 stands in the middle of 2 distinct primes. 1 is here prime q.e.d.

Verify with http://unsolvedproblems.org/S20.pdf A proof of Goldbach and de Polignac conjectures. Jamel Ghanouchi. 6 Rue Khansa 2070 Marsa

(n-a) and (a-n) is here the primal radius in Jamels proof

M.B-Sievert — Preceding unsigned comment added by M-B-Sievert (talk • contribs) 07:25, 2 June 2012 (UTC)

Ahmad Sabihi

The Author has published a 47-page journal paper in 2010 namely:The novel researches toward the proof of the Goldbach's conjecture by the novel functions, the novel conjecture, the Riemann zeta function, and the novel experimental computations,Bull. Allahabad Math. Soc. 25, No. 1, 77-123 (2010). This paper indexes in MathScinet and Zentralblatt-Math.The Author presents 5 new functions on the even integer sets. Then based on them, He makes a conjecture namely Sabihi's conjecture. He claims that if his own conjecture is proved for evens greater than 120 to infinity,then Goldbach's conjecture will be proved for all even integers. Author also gives the experimental evidence by computer analysis that would indicate the Sabihi's conjecture is true for the even numbers 120 to 100,000 and might be true for the evens greater than 100,000 to infinity. — Preceding unsigned comment added by Sabgold (talk • contribs) 12:49, 7 October 2012 (UTC)

First of all, when you create a new section in the talk page, please put it at the end. The best way to do this correctly is to use the button "New section" at the top of the page. Secondly, please sign your posts by writing four tildes (~~~~) at the end or by using the button with apencil at the top of the edit window.

About the content of your post: An encyclopedia, like Wikipedia, can not report all the papers published worldwide. For scientific results, as it is the case here, it may be cited only if external and reliable sources may attest that 1- the result is correct, and 2- it is sufficiently notable. This paper has been published in a non notable journal, whose reliability of the referee process (if any) is unknown. It has never been cited in any other paper. Thus there is no third party source attesting correctness and importance of Sabihi's work. If someday this work appears to be correct and important, it could be the case to mention it. But it is not yet the case (see WP:CRYSTAL). --D.Lazard (talk) 13:22, 7 October 2012 (UTC)

The relevant policies here are tge policy against original research, especially the part forbidding us to rely on primary sources such as the source under discussion, and the policy on maintaining a neutral point of view, specifically the part that says we should give weight to different viewpoints in proportion to their prevalence in reliable sources. Many thousands of reliable papers have been written on Goldbach's conjecture, many of which count as secondary sources. But none of these mention the work under discussion. Sławomir Biały (talk) 13:44, 7 October 2012 (UTC)

Thank you for your clarifications D.Lazard. In responding to you and defending on my work, I would like to tell you that firstly this paper has been completely refereed for about 1.5 years with twice revising it, secondly this journal is a notable one due to its editor is Professor H.M. Srivastava who has published more than 1000 papers in mathematical journals as well as he is the editor of more than 100 journals around the world. Thirdly,as I mentioned at the previous paragraph, the paper is indexed by two great mathematical community MathScinet(USA) and Zentralblatt-Math(Europe)and several well-known libraries as Tsinghua University,and Mathematical institution in Serbia (Belgrad) and etc. Please do not judge about content of an academic paper, before reviewing it, specially if great mathematicians are the editors of publishing that paper. The Allahabad Mathematical Society is one of the well-known scientific societies registered by American Mathematical Society and there are several great mathematicians as members of the council,editorial boards, and committee for publications.--109.122.212.14 (talk) 17:44, 7 October 2012 (UTC)

Dear Lazard,I'd like to send you the abstract and conclusions of the paper to be reviewed by you. Also, I should mention that I myself completed a Ph.D. in mathematics (number theory) under supervision of Prof. David Rohrlich in 1996. My Ph.D. dissertation is on the Goldbach conjecture by title: A proposal proof for Goldbach's conjecture. I have discussed it with many mathematicians expert in number theory and then after about 14 years I could finally publish it in a journal. Therefore, this paper has passed a long period of discussing as well as explanations given at the above paragraph. Also, I have given many lectures in mathematics and number theory as a professor in some universities. Hence, you should trust to my paper to be cited in Wikipedia. Please kindly let me so I cite it in Goldbach' conjecture Wikipedia's page. A citation of it, may help to excite people to track and propagate it worldwide. Such a paper of many volume of data and new ideas is rarely published. Also, I have looked at some Wikipedia's pages and found some articles, which cited in it but none published anywhere. For example in arXiv:math. But my paper has been published and indexed. Also, anybody haven't reported problem or conflict with.--Sabgold (talk) 11:24, 10 October 2012 (UTC)

The last preceding post will be copied to User talk:Sabgold, which is the right place for such a discussion.--D.Lazard (talk) 12:43, 10 October 2012 (UTC)

Thank you for your reply D.Lazard. I am not yet convinced with your clarifications nor challenge with you or other editors. It seems you discriminate against me, because for example, same Wikipedia's page of Goldbach's conjecture cites Nicola Fragnito's work on Goldbach's conjecture published in Jp Journal of Algebra and Number Theory in 2011. Also, his another paper namely “EULER BRICK” in the Pioneer Journal of Algebra, Number Theory and its Applications" published in 2011, is cited in the Wikipedia. None of these papers are neither cited anywhere nor published in notable journals, but they are cited in the Wiki pages!!!.Why is such?! In my opinion, you and other European-American editors do discriminate against people of the other nationalities. This is not a way of interaction with the world. Please let the rest people of the world so can progress their works and attempts!!!. In any case, I will report and complain to the official ministration of the Wikipedia's company, your prejudicial manner with respect to me.Sabgold (talk) 00:16, 11 October 2012 (UTC)

Fragnito's paper has been removed 3 times from this article and there is a section about it in this talk page. Other spam citations of the same author have been inserted by an IP user and removed in Collatz conjecture and Twin prime. Thank you for pointing us that problem in Euler brick. It has been solved now. D.Lazard (talk) 03:16, 11 October 2012 (UTC)

Notation for Goldbach partition

What does the notation in the section Goldbach's conjecture#Goldbach number mean? For example 2(5) = 10 = 3 + 7 = 5 + 5. Is that the partition function? If so, it seems that 2(n) denotes a partition of n into two summands, but this is explained neither in this article, nor in partition (number theory), so I am not sure. -- Toshio Yamaguchi 13:38, 5 January 2013 (UTC)

It appears from the context that 2(n) is simply intended to mean 2×n, but I don't see a reason for the notation or for writing anything at all there. It was added in [9] with edit summary "tweaked samples for clarity", but it looked clearer to me before. PrimeHunter (talk) 15:29, 5 January 2013 (UTC)

I took the notation out. The article specifically has been stating that a Goldbach number is even before, so I think that makes it easier to understand. -- Toshio Yamaguchi 15:50, 5 January 2013 (UTC)

Conjecture proven (maybe)?

I just found this paper on arXiv. I don't know whether it has been accepted by a mathematical journal though or whether this is notable at all. It might just be another one of those purported proofs of a famous mathematical problem. -- Toshio Yamaguchi 09:30, 9 January 2013 (UTC)

Well, I found this where the source claims it will be published in International Journal of Pure and Applied Mathematics. So lets wait and see if one of them survives peer-review.... -- Toshio Yamaguchi 10:00, 9 January 2013 (UTC)

The arXiv paper consists mainly in restating and reproving many elementary well known facts. This makes difficult to find where one can find the alleged proof in the paper. It is said, in the introduction that the proof relies on a "new criterion", which consists in the brute force algorithm for testing Goldbach conjecture. Thus it is highly improbable that the paper contains a correct proof. D.Lazard (talk) 10:54, 9 January 2013 (UTC)

In the introduction it is stated that the Goldbach conjecture follows as a corollary from Theorem 2.20. I haven't looked into this deeper so far though. -- Toshio Yamaguchi 12:31, 9 January 2013 (UTC)

Don't get too excited. Alleged Goldbach proofs are so common that the Prime Pages author professor Chris Caldwell asks people to not post them in the description at http://tech.groups.yahoo.com/group/primenumbers/. His Crackpot index awards 20 points for a purported proof. PrimeHunter (talk) 18:03, 9 January 2013 (UTC)

Okay, I get a score of 25 points for that one then (5 for considering 1 to be prime). -- Toshio Yamaguchi 20:43, 9 January 2013 (UTC)

Related to the Bertrand-Chebyshev theorem ?

...which states that there's at least one prime inbetween a number and its double ? — 79.113.221.97 (talk) 22:48, 19 March 2013 (UTC)

Goldbach's conjecture implies the weaker formulation of the Bertrand-Chebyshev theorem. I don't think it's worth mentioning in the article, considering the theorem was already proved by other means in 1850 while Goldbach's conjecture remains open. PrimeHunter (talk) 00:57, 20 March 2013 (UTC)

Harald Helfgott's proof of the weak Goldbach conjection

I was going to remove this, but I think there's enough here that it's worth a note in the article. It's not just an arVix page; we have http://www.truthiscool.com/prime-numbers-the-271-year-old-puzzle-resolved reporting on it, and we have https://plus.google.com/u/0/114134834346472219368/posts/8qpSYNZFbzC a Fields Medalist asserting it to be true. It's not strong enough that we should trumpet it, but it's a solid claim from reliable sources. Terence Tao asserts that Harald Helfgott's proof is good, when both of those people are professional mathematicians with several awards, is good enough for me.--Prosfilaes (talk) 19:24, 14 May 2013 (UTC)

I don't think this that is currently enough, nvm that www.truthiscool.com nor the arxiv-preprint are acceptable sources at this point. Tao's blog entry about certainly provides for more relevance but on its doesn't necessarily justify mentioning it either. Note the paper was just published 2 days ago and afaik there hasn't been any thorough review yet. If a paper mistake/flaw in that paper shows it might not even worth to mentioned in the article and there is no rush to include this, i.e. we should wait for the dust to settle and then include it.--Kmhkmh (talk) 19:54, 14 May 2013 (UTC)

If something changes, then it changes; it's no harder to take out then it was to put in. There's no need for us to wait for the dust to settle; we don't wait for 2014 to create the 2013 Atlantic hurricane season.--Prosfilaes (talk) 20:02, 14 May 2013 (UTC)

Now we have a New Scientist cite: http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html --Prosfilaes (talk) 20:07, 14 May 2013 (UTC)

Because we will have an article on the hurricane season either way. However whether we will have information on this proof in the article depends entirely of the results of the review. In otherwords your analogy is completely false.--Kmhkmh (talk) 20:36, 14 May 2013 (UTC)

We will have an article on the hurricane season either way, but we will have to change wind-speeds and pressures for all the hurricanes when we get the post-season data.--Prosfilaes (talk) 10:23, 15 May 2013 (UTC)

I think it should be removed for a couple of reasons. The way it's stated appears to give credit to Helfgott for proving the conjecture, but this is inaccurate because it has been already known that the conjecture is true for sufficiently large odd integers. Also, while it's OK to mention important results even if they're still being verified, I'm not sure that verifying Goldbach in a finite range meets this criteria...Spwood2 (talk) 05:23, 15 May 2013 (UTC)

You say that it's known to be true for sufficiently large odd integers, but the article does not say that, nor do the links that mention this proof. And a proof for all integers is still a proof for all integers; a proof for sufficiently large odd integers leaves out (literally) untold numbers of intervening integers.--Prosfilaes (talk) 10:23, 15 May 2013 (UTC)

It's not "(literally) untold numbers of intervening integer" because we were actually told the range of the possible exceptions explicitly by Vinogradov and others! In any case, yes, you're right, the article does not say the conjecture is true fo sufficiently large odd integers, but that's because this article is concerned with the Goldbach conjecture whereas the result being discussed is about the weak Goldbach conjecture, for which there is a separate article on Wiki! Note that this is yet a third reason why this should be removed: If anywhere, it belongs to the weak goldbach conjecture article and not here! Spwood2 (talk) 12:42, 15 May 2013 (UTC)

Just to clarify, the new work of Helfgott does two things: (1) prove numerically that the conjecture is true for numbers up to ~10^30 (improving on previous results by several orders of magnitude); and (2) prove theoretically that the conjecture is true for numbers above ~10^30 (improving on previous results, proving numbers above ~e^3100 by countless orders of magnitude). Together it obviously proves the conjecture for all natural numbers. To say that this is a small improvement over previous knowledge/proofs in this area is ridiculous. 93.173.144.154 (talk) 11:53, 15 May 2013 (UTC)

OK, I guess the reason this is being considered an important improvement is because the number ~e^3100 is very large, which I agree it is. Spwood2 (talk) 12:22, 15 May 2013 (UTC)

I would like to see this left in. It's made a stir and received some coverage. I also know Harald, and he's definitely a hot shot in this area, so this represents a serious breakthrough, even should the proof turn out to have a flaw. The same obviously can't be said of the dozens of claimed "proofs" that appear each year. Sławomir Biały (talk) 12:54, 15 May 2013 (UTC)

Goldbach Conjecture <=> Binomial

1,a,b ∈ ℙ, a>b , n>1 ∈ ℕ

2n=(a²-b²)/(a-b) => 2n=(a+b)

4 = (3²-1²)/(3-1) = (3+1) = 2*2

6 = (5²-1²)/(5-1) = (5+1) = 2*3

8 = (5²-3²)/(5-3) = (5+3) = 2*4 — Preceding unsigned comment added by 84.135.36.22 (talk) 22:04, 27 October 2013 (UTC)

Is there a point to this? PrimeHunter (talk) 22:24, 27 October 2013 (UTC)

The German, possibly in Munich, seems to think that he has found a proof of the conjecture. He has overlooked the fact that 1 is not now seen to be a prime. He does not apply his theory explicitly to higher even numbers, like 10. — Preceding unsigned comment added by 86.183.72.219 (talk) 10:14, 7 March 2014 (UTC)

10 = (7²-3²)/(7-3)=(7+3)=2*5 100= (97²-3²)/(97-3)=(97+3)=2*50 So if Goldbach holds its binominal if its binominal Goldbach holds. (1 is here prime) — Preceding unsigned comment added by 84.135.41.43 (talk) 08:29, 27 March 2014 (UTC)

For any numbers a≠b, (a²-b²)/(a-b) = a+b. So if a number can be written as a+b then it can be written as (a²-b²)/(a-b). This does not help solve Goldbach's conjecture and is not worth adding to the article. a+b can be written in infinitely many other ways and I don't see a reason to mention this one. PrimeHunter (talk) 13:37, 27 March 2014 (UTC)

Every n stands between two odd primes p and q

1 is here prime (Goldbach)

Bertrands postulate is true

p*q = n²-a² = (n+a)(n-a)

p+q = (n+a)+(n-a) = 2n

Goldbach Conjecture is true — Preceding unsigned comment added by 84.135.35.142 (talk) 11:14, 13 May 2014 (UTC)

If tightened up a little, you are showing that the sum of two odd primes is an even number. This shows nothing about every even n. If you think it does then it's original research and doesn't belong in Wikipedia. PrimeHunter (talk) 11:27, 13 May 2014 (UTC)

3325581707333960528

From the article: One record from this search: 3325581707333960528 is the smallest number which has no Goldbach partition with a prime below 9781.

I deleted it once on the grounds that they were arbitrary numbers, and still in this form I object. Perhaps if it says something interesting about the Goldbach conjecture and is phrased that way, it will be more interesting, but right now it's still two arbitrary numbers.--Prosfilaes (talk) 05:44, 30 April 2014 (UTC)

I think it says something interesting that 9781 is so small. It is only the 1206th out of around 4×10¹⁶ primes below 3325581707333960528 / 2, and the rest are never needed to find a partner with the right sum. PrimeHunter (talk) 12:50, 13 May 2014 (UTC)

"odd primes"

The first sentence of the definition section states: "A Goldbach number is a positive integer that can be expressed as the sum of two odd primes." - I find it distracting to have the strange redundancy of stating that these integers can only be from the set of primes that are odd, since that is provably the set of all primes known and unknown. Absent a formalistic reason for stating it that way, would it not be clearer to simply state it as "the sum of two primes"?

Xuancris (talk) 17:00, 27 March 2015 (UTC)

2 is an even prime (the only one and therefore sometimes called "the oddest prime"). 4 = 2+2 can only be expressed as the sum of even primes and is therefore not a Goldbach number. PrimeHunter (talk) 17:20, 27 March 2015 (UTC)

As "even" precisely means "divisible by 2", I don't see the point in attributing to 2 such a special statute. At least not for this reason only. 3 is also a very special prime, as it is the only one that is divisible by 3.Sapphorain (talk) 22:12, 27 March 2015 (UTC)

Calling 2 "the oddest prime" is a common joke playing on two different meanings of "odd". I'm not saying the article should claim 2 is a very special prime but the source [10] defines a Goldbach number as a sum of two odd primes so we should do the same as long as that source is used. The term "odd primes" is actually used in many situations where something doesn't apply to the prime 2. The set of primes not divisible by 3 is very rarely considered, but the set neither divisible by 2 nor 3 is sometimes considered. PrimeHunter (talk) 22:40, 27 March 2015 (UTC)

What is the first figure meant to show?

What is the explanation for the first figure? The blue and pink pyramid looks pretty, but does it convey any information? Norman21 (talk) 10:04, 3 May 2015 (UTC)

That illustration is pretty obtuse, now you mention it. I just tried to write a better caption, and tied myself up in linguistic knots. The colors have no intrinsic meaning, and it's hard to explain why there are no circles marking many of the intersections.

It's a nice illustration for people who already understand the concept, but I think a table would be more informative for the reader who comes not understanding the conjecture. - DavidWBrooks (talk) 11:42, 3 May 2015 (UTC)

 Fixed. I have rewritten the caption for explaining the figure. I hope it is clearer. D.Lazard (talk) 12:22, 3 May 2015 (UTC)

Now I understand! Thank you!! Norman21 (talk) 17:39, 3 May 2015 (UTC)

ordered vs. unordered partition

There is some mixup in the section on "heuristic justification" that should be fixed. In the text concerning Hardy-Littlewood's 1923 Acta Math. paper it is stated that the counting of prime tuples is being done with p_1 =< p_2 =< .... =< p_c (which means that, e.g., wehn c=2, given the two primes 3 and 5, only the pair (3,5) is counted as a partition of 8 but not the pair (5,3)), and eventually a formula is given for the special case c=2. However, that formula is for when, e.g. 8 = 5+3 and 8 = 3+5 are counted as two distinct ways of writing 8 as a sum of two primes. (In the words of Hardy and Littlewood, the counting is done with attention being paid to order.) By contrast, the scatter plots of Goldbach's comet to the right of the text in question are indeed for when counting is done without paying attention to order. So the asymptotic formula stated gives a result which is a factor 2 too high when compared to the plot. The upshot is, drop the factor of 2 in the formula (twice) to make it consistent with what's written, and with the scatter plots. -MKHKMkhkATwiki (talk) 13:46, 20 July 2015 (UTC)

Concern about copyright vio

I notice that the Origins section of this article appears to be copied verbatim from the book Unsolved Problems in Mathematics, by Fredrick Kennard. See here on Google Books:

https://books.google.com/books?id=OaNsCQAAQBAJ&pg=PA66&lpg=PA66&dq=goldbach%27s+marginal+conjecture&source=bl&ots=P8b8d_pRXI&sig=UudAR8n30AulAPswC8QlfSWzs_c&hl=en&sa=X&ved=0CEEQ6AEwBWoVChMI2JS4o7_ixwIVA8-ACh1ThAYG#v=onepage&q=goldbach's%20marginal%20conjecture&f=false

Idempotent (talk) 13:27, 6 September 2015 (UTC)

It seems much more likely that the copying went the other way. Sławomir
Biały 15:14, 6 September 2015 (UTC)

Yes, exactly: since this book was published in February 2015, and the Section "Origin" in the article has been as it is much before that date, it seems it is the other way around: Mr Kennard has copied the wikipedia article (which he is perfectly entitled to do).Sapphorain (talk) 15:22, 6 September 2015 (UTC)

Thanks! Sorry, I should have thought to check the dates myself, but somehow I overlooked the possibility that copying could go this direction! OK, good to know! Idempotent (talk) 15:52, 6 September 2015 (UTC)

The whole book is copied from Wikipedia. Chapter 3 "Text and image sources, contributors, and licenses" links to the Wikipedia articles and lists their contributors. There are hundreds of thousands of books like this, if not millions by now. I once started a list at User:PrimeHunter/Alphascript Publishing sells free articles as expensive books with the idea that potential book buyers who search a title on the Internet would find the page and see what they pay for, but I quickly gave up keeping track. PrimeHunter (talk) 18:27, 6 September 2015 (UTC)

I must say it is a great relief to me to verify that Mr F. Kennard is credited with zero publication by MathSciNet. I would be even more relieved if I could make sure that he is not a mathematician, in any sense of the term. Sapphorain (talk) 19:57, 6 September 2015 (UTC)

And the ones I've seen have a very high price on them. On some on Amazon I've put in the reviews that it is available for free on Wikipedia (and likely more up to date). Bubba73 ^{You talkin' to me?} 23:13, 6 September 2015 (UTC)

is 4 a Goldbach number?

It is confusing to have a definition of G's conjecture that includes 4 ("Every even integer greater than 2 ...") but then have the very next section on "Goldbach number" define them in a way that does not include 4 ("sum of two odd primes" since 1 isn't usually considered a prime). The recent back-and-forth in whether to include 4 in the second section demonstrates the confusion. At the very least, that section needs a discussion as to why "Goldbach number" is defined in a way to skip 4. - DavidWBrooks (talk) 12:13, 15 January 2016 (UTC)

 Done Bill Cherowitzo (talk) 04:31, 16 January 2016 (UTC)

Supposed proof

This edit claims that the conjecture has been proven in a paper in the IOSR Journal of Mathematics vol12 issue6 version 1 [11]

I've tracked down the exact source to http://www.iosrjournals.org/iosr-jm/papers/Vol12-issue6/Version-1/Q120601113117.pdf

The IOSR is the International Organization of Scientific Research, and it claims to publish mathematical papers in less than one month. I'm not an expert on this, but that doesn't sound like a reliable scientific journal source for a claim to teh solution of Goldbach's conjecture. Meters (talk) 02:35, 6 January 2017 (UTC)

This journal is neither reviewed, nor even indexed, by MathSciNet or Zbl. This means it is in no way reliable. Sapphorain (talk) 06:07, 6 January 2017 (UTC)

Thanks for the confirmation. Meters (talk) 06:10, 6 January 2017 (UTC)

I had a look on the article. The claimed proof cannot be correct, as it uses only very elementary tools, well known by every mathematician that has worked on the subject (including Goldbach himself). If such an elementary proof would exist, it would have been found for a long time. D.Lazard (talk) 09:42, 6 January 2017 (UTC)

I was certain the proof was bogus but read it anyway out of curiosity. It's trivially false as expected. It only uses high school algebra and makes high school level errors. It basically tries to prove:

For even n and any a > 1 which doesn't divide n, there exists prime p < n such that n = p + m, where a does not divide m.

Even if this proof was correct it would lead nowhere. The paper totally misses that each a may lead to a different (p, m) pair, and there may be no m which simultaneously avoids all a as factor. PrimeHunter (talk) 16:50, 6 January 2017 (UTC)

It's vaguely similar to Euclid's theorem.--♦IanMacM♦ ^{(talk to me)} 17:11, 6 January 2017 (UTC)

Prenex normal form

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

---

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)

---

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)

---

• Why have separate a,b and c,d? One can reuse a,b. 114.243.152.197 (talk) 16:51, 25 April 2018 (UTC)

Solved?

There are claims that a Guinean, Sambégou Diallo, has solved the conjecture. http://culture-kamite.com/ibrahima-sambegou-diallo-le-guineen-qui-a-solutionne-un-probleme-de-math-vieux-de-270-ans/ 114.243.152.197 (talk) 16:46, 25 April 2018 (UTC)

The claim dates back at least four years (http://africancreationenergy.blogspot.com/2014/01/a-guinean-solves-270-years-old.html) but I don't see any confirmation anywhere. - DavidWBrooks (talk) 16:56, 25 April 2018 (UTC)

This is similar to the claim that the Nigerian professor Opeyemi Enoch solved the Riemann hypothesis in 2015.[12] Neither claim has been verified by peer review.--♦IanMacM♦ ^{(talk to me)} 17:03, 25 April 2018 (UTC)

Sambégou Diallo's February 2013 proof can be read here. It's in French, sorry Anglophones. It is on vixra.org (arxiv spelled backwards), which describes itself as "An alternative archive of 23384 e-prints in Science and Mathematics serving the whole scientific community" (translation, a place where people can submit any academic document). There is another proof of Goldbach's conjecture from February 2017 on the site here, this time in English.--♦IanMacM♦ ^{(talk to me)} 07:38, 27 April 2018 (UTC)

Proof?

Someone is claiming to have a proof here: https://vixra.org/pdf/1702.0150v1.pdf. Not sure I fully agree that it's a proof, though. Seems to me to assume that the weaker conjecture means we can choose any odd prime to represent p_3. p_3 and k depend on each other, assuming we don't know if the Goldbach conjecture is true. Heymid (contribs) 18:20, 10 April 2020 (UTC)

It's on Vixra, ergo you should ignore it. --JBL (talk) 18:55, 10 April 2020 (UTC)

I see. There are no requirements for publication. Looked like a nonsensical proof to me anyway. You're right, somebody's just trying to sound smart without actually coming up with any real arguments. I see no reason to suggest the strong conjecture follows from the weaker one. Heymid (contribs) 18:59, 10 April 2020 (UTC)

See my post from 27 April 2018 above, which mentions Marshall's February 2017 proof. This is now over three years ago and no-one seems to think that Goldbach's conjecture has been proved.--♦IanMacM♦ ^{(talk to me)} 19:37, 10 April 2020 (UTC)

The paper you referenced was republished under an account on Vixra using his middle-name and and the second version on the website, https://vixra.org/abs/1702.0150, was rebutted in the comments with a separate publishing. I checked the rebuttal of the second publishing in the comments and it checks out, which explains why Stephen re-published a third and fourth version of the proof you've mentioned because he feels he cannot pursue the other route of proof. I took the time to thoroughly reply to Stephen in the comment section and that will help anyone else who finds this talk section and is interested in investigating why Stephen Michael Marshall's proof methods are incorrect. Best wishes. I'm not going to sign this. — Preceding unsigned comment added by 107.136.154.2 (talk) 11:38, 26 May 2020 (UTC)

infinite prime?

Question: Is the reason the conjecture cannot be proved because it does not make working sense when one of the primes is infinite? 82.11.40.109 (talk) 18:36, 20 August 2020 (UTC)

Put another way: If the even number is infinite then at least one of the primes has to be infinite. Is there such a beast as an infinite prime? If not then that's one case that doesn't work. Stuffed Cat (talk) 19:02, 20 August 2020 (UTC)

That is not correct. There are an infinite number of primes - there is no largest prime - but all of them are finite. - DavidWBrooks (talk) 19:06, 20 August 2020 (UTC)

Okay so if all primes are finite then there are no two primes that will add up to an even infinity. So if the conjecture does not work for infinity then it isn't globally true. It might work just for non-infinite even numbers - nobody has found an even number that don't work (yet) - but how do you prove that it works leaving ut infinity? Pi R squared and Pythagoras work and makes a kind of sense if one of the dimensions was infinite but Goldbach just doesn't - not for infinity anyway. Stuffed Cat (talk) 22:30, 20 August 2020 (UTC)

You should drop this. You apparently don't understand infinity, and this is getting into WP:NOTAFORUM. Meters (talk) 22:49, 20 August 2020 (UTC)

Discussion / argument moved to another place. Stuffed Cat (talk) 23:03, 20 August 2020 (UTC)

Lead

Current lead reads:

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even integer greater than 2 is the sum of two primes.

But WP:LEAD tells us that "the first sentence should tell the nonspecialist reader what, or who, the subject is." and later "...the first sentence is usually a definition". The current text does not do that. Instead, it starts with a statement of notability. This lead also uses three technical terms that may not be familiar to the non-specialist: "number theory", "integer", and "prime". I propose to start with what it is before talking about its significance, like this, and to use less technical terms:

In mathematics, Goldbach's conjecture is that every even whole number greater than 2 is the sum of two prime numbers. It is one of the oldest and best-known unsolved problems in number theory and all of mathematics.

This explicitly says "prime number" rather than "prime" (which is somewhat jargony) and "whole number" rather than "integer" to ensure that it's understandable to a large audience. Of course the current version is perfectly understandable to a college math major, but the conjecture itself can be understood by a high school student, and we should use language that the high school student will understand. Comments? --Macrakis (talk) 20:16, 24 December 2020 (UTC)

"...the first sentence is usually a definition". Usually, but not always. :) I am not an expert on subtleties of the Wikipedia style, but in my feeling the current order of the sentences is good, but maybe it would be better to change words as you propose: "prime number" instead of "prime" and "whole number" instead of "integer". --D.M. from Ukraine (talk) 20:39, 24 December 2020 (UTC)

(edit conflict) With respect to the order of the sentences: the Goldbach conjecture is an unsolved problem in number theory. To compare with the first two examples I thought of, the P versus NP problem is "a major unsolved problem in computer science" and the Poincare conjecture is "a theorem about the characterization of the 3-sphere". With respect to the wording of the conjecture itself (in what is presently the second sentence): I agree with you that those small wording changes would make it more accessible. --JBL (talk) 20:45, 24 December 2020 (UTC)

Since that was three quick votes in favor of the wording changes, I have implemented them. --JBL (talk) 20:47, 24 December 2020 (UTC)

Saying that something is "an unsolved problem" doesn't tell the reader what it is, only what class of things it belongs to. I agree that P versus NP problem and Pi start in a similar way, and I think they are equally problematic. The lead sentence to e (mathematical constant) is somewhat better. The lead of Poincare conjecture is much better, because it says what the theorem is about. Since the full statement is dense and technical, the "characterization of the 3-sphere" is probably about as good a non-specialist description as we're going to get.

I will be interested to hear what other editors think. --Macrakis (talk) 21:32, 24 December 2020 (UTC)

I also agree it would be good to hear what other editors think, but (since you've directly responded to me) I can't help wondering what you think it means to say what something is. Like, consider the article Toyota Camry: it currently begins "The Toyota Camry is an automobile sold internationally by the Japanese manufacturer Toyota since 1982...." It tells you what kind of thing a Toyota Camry is -- what class of things it belongs to. Or take The Lord of the Rings: "The Lord of the Rings is an epic high fantasy novel...." How do you think these articles should begin? --JBL (talk) 22:48, 24 December 2020 (UTC)

The reason why it is famous is because it is an "unsolved problem". In that sense that is what it is. Exactly what kind of unsolved problem, for the general reader, is secondary. Paul August ☎ 01:16, 25 December 2020 (UTC)

IMO, the order of the two first sentence must not be changed. The statement of the conjecture can be understood by almost everybody, but does not allows distinguishing it from many other conjectures and theorems of elementary number theory. It is its historical importance that makes it different. This is the main reason for keeping the order of the sentences. D.Lazard (talk) 17:17, 25 December 2020 (UTC)

I didn't express myself very well. Let me try again.

The best-known or most notable characteristic of a topic is not necessarily its defining characteristic, which is what the Manual of Style tells us to start with. The MOS tells us not to start articles like this:

• The Edsel car is a notorious example of a failed product launch by a major corporation.
• The Enron corporation is a symbol of willful corporate fraud and corruption.
• The coelacanth fishes are living fossils which were thought to have become extinct 66 million years ago.
• Jeff Bezos is the world's richest person.
• The element mercury is the only metal that is liquid at room temperature.
• Herodotus was the first historian to write systematic history.

Each of these examples mentions the category of thing the subject belongs to (car, corporation, fish, person, element, historian) but does not characterize its specifically. Instead, it talks about its importance or best-known characteristic. The articles do mention those characteristics later in the lead paragraph (or soon thereafter), but not in the first sentence, which is supposed to define the topic, not tell the reader its importance (historical or otherwise).

If I were King of Wikipedia, I might say that those are better lead sentences than the current ones, but a long time ago, Wikipedia settled on a house style where we first characterize the topic, and only then comment on why it is notable. --Macrakis (talk) 16:18, 26 December 2020 (UTC)

Yes sure but this has nothing to do with the edit you made (and that I infer you are still defending). Here's how Edsel actually begins: "Edsel is a brand of automobile marketed by the Ford Motor Company ...." This is precisely parallel to "The Goldbach conjecture is an open problem in the mathematical field of number theory ...." It is not even slightly parallel to "The Goldbach conjecture is that every even number larger than 2 is a sum of two primes", or whatever. If you had rewritten the lead paragraph to say "Goldbach's conjecture is an unsolved problems in the mathematical field of number theory. The conjecture states that every even whole number greater than 2 is the sum of two prime numbers. It is one of the oldest and best-known open questions in all of mathematics." then we would be having a very different conversation! --JBL (talk) 16:36, 26 December 2020 (UTC)

Clearly car brands can't be characterized as concisely as mathematical statements. But "one of the oldest and best-known unsolved problems" is awfully vague. At least in the Edsel case the first sentence mentions the manufacturer and the dates. The equivalent for Goldbach might be "Goldbach's conjecture is a statement about the additive decomposition of prime numbers which has neither been proven nor disproven". But at that point, we might as well state the thing.... --Macrakis (talk) 18:00, 26 December 2020 (UTC)

Sorry. It is not about the additive decomposition of prime numbers, but into prime numbers. --Macrakis (talk) 23:43, 27 December 2020 (UTC)

Episode of Lewis

Re this edit: It has been added before, but the sourcing and plot summary on IMDb are not very good.[13] "Reputation" is the pilot episode of Lewis (TV series) from 2006 and I have watched it to see what it is about (plot spoiler ahead). Lewis is called in to investigate the death of an Oxford mathematics student, and eventually discovers that the password to get into the computer of Danny Griffon is 8128, because he is fascinated with perfect numbers. On his computer there is an apparent proof of Goldbach's conjecture, but this later turns out to be incorrect. Out of the entire one hour 33 minutes run time, very little is about Goldbach's conjecture and it is only a passing reference in a convoluted murder mystery. So it isn't all that notable anyway.--♦IanMacM♦ ^{(talk to me)} 16:37, 29 January 2021 (UTC)

I agree. Famous math problems get mentioned reasonably often in popular culture; "it got mentioned" without anything substantive to say (as would be provided by a quality secondary source) is pointless. --JBL (talk) 16:45, 29 January 2021 (UTC)

There are some screenshots from the TV episode here. The plot also features a lock with the combination 496, which is also a perfect number.--♦IanMacM♦ ^{(talk to me)} 16:57, 29 January 2021 (UTC)

Twin prime constant naming

Why is the twin prime constant $C_{2}$ name changed to $\Pi _{2}$? Is there some historical context?

Also, I added a link and now:

• Hardy–Littlewood prime tuple conjecture
• Hardy–Littlewood's twin prime constant

have the same destination. My new link comes first, so should I delete the second link?

Darcourse (talk) 00:18, 15 July 2021 (UTC)

"every even integer greater than 4,208 is the sum of two twin primes."

To me that line suggests a pair of twin primes would form said even integer, (n/2 and n/2 +/- 2), but that's not the case. 2 primes, usually(? almost always?) from 2 different sets of twin primes, would sum to that number.

I'd edit it myself, but I wanted to confirm with actual math experts first.--Skintigh (talk) 18:46, 23 August 2021 (UTC)

Not my area, but the line does not say that the primes need to be from the same pair of twin primes, and the cited source https://oeis.org/A007534/a007534.pdf confirms that. Even if we do clarify that the twins are not necessarily from the same pair there is no treason to specify how frequently this happens. Meters (talk) 20:49, 23 August 2021 (UTC)

The previous line at Goldbach's conjecture#Twin primes defines a twin prime as an individual number:

A twin prime is a prime number p such either p − 2 or p + 2 (or both) is prime.

I see no need to state that the two twin primes can be from different pairs. It seems obvious and I haven't seen a source doing it. OEIS:A007534 even uses the term "pair" about them: "Even numbers that are not the sum of a pair of twin primes." PrimeHunter (talk) 22:51, 23 August 2021 (UTC)

Short description

Re this edit: I really don't think this is an improvement on the previous wording, what do others think?--♦IanMacM♦ ^{(talk to me)} 17:29, 22 February 2022 (UTC)

The previous description was Conjecture that every even integer > 2 is the sum of two primes. It is too long. If truncated at 40 characters, the reader does know the most important fact, namely that is is about the sum of two primes. What such a reader sees is a repetition of a technical word of the title (conjecture), another word that may be technical for a layman (integer instead of number) and a minor technicality (>2).

On the other hand, you have reverted, without giving any other reason than your own preference, the following short descriptions: Are even numbers sums of two primes? and On even numbers as sums of two primes. The former has also been reverted by another user with the edit summary "A conjecture is not a question"; this is controversial, as a conjecture is ultimately the question whether a statement is true. Personally I prefer the short description with a question mark, as it is more accurate, but both follow the guidelines WP:SDSHORT and WP:SDNOTDEF as well as the implicit consensus at WT:WikiProject Mathematics/Archive/2022/Jan#Short Description Guidelines. D.Lazard (talk) 18:11, 22 February 2022 (UTC)

No Highway in the Sky

Re this edit: It's true that James Stewart refers to Goldbach's conjecture in the 1951 film No Highway in the Sky.[14] He says that "it has been verified to 10,000 but never proved". Since it is only a passing reference and is not a key part of the plot, it does not meet WP:POPCULTURE guidelines. ♦IanMacM♦ ^{(talk to me)} 11:04, 7 July 2022 (UTC)

Short description

Regarding this diff, @Ianmacm: per WP:SHORTDESC, short descriptions should be under 40 characters and should help readers identify what the topic is about in search results. "Unsolved problem in mathematics" does this better than "Conjecture that every even integer > 2 is the sum of two primes", particularly because searchers may not know what a conjecture even is. Sandstein 08:40, 19 December 2022 (UTC)

Note also that short descriptions are not definitions, but explanations of the scope of a page. That's an important difference. Sandstein 08:42, 19 December 2022 (UTC)

To use less than 40 characters is recommended, not compulsory. "Unsolved problem in mathematics" is so vague it's useless. --Sapphorain (talk) 09:18, 19 December 2022 (UTC)

A short description should provide:

• a very brief indication of the field covered by the article
• a short descriptive annotation
• a disambiguation in searches, especially to distinguish the subject from similarly titled subjects in different fields

The reverted edit was too short and vague to give an idea of what the article covers, and could have been applied to any unsolved problem in mathematics.--♦IanMacM♦ ^{(talk to me)} 09:42, 19 December 2022 (UTC)

I agree with the revert and the given reasons. However, the restored version contains too many useless words, especially at the beginning. It begins with "Conjecture that every", that is, a partial repetition of the title and two words that do not carry any meaning by themselves. So, I have replaced the short description with "Even integers as sums of two primes". D.Lazard (talk) 10:26, 19 December 2022 (UTC)

Semi-protected edit request on 14 January 2023

The paragraphs:

---

The number of representations is about n ln ⁡ n n\ln n, from 2 n = p + c {\displaystyle 2n=p+c} and the Prime Number Theorem. If each c is composite, then it must have a prime factor less than or equal to the square root of 2 n 2n, by the method outlined in trial division.

This leads to an expectation of n ln ⁡ n 2 n = n 2 ln ⁡ n {\displaystyle {\frac {n\ln n}{\sqrt {2n}={\sqrt {\frac {n}{2}}}\ln n} representations.

---

are complete crap. Delete. The sentence before:

---

Goldbach's comet also suggests that there are tight upper and lower bounds on the number of representatives, and that the modulo 6 of 2n plays a part in the number of representations.

---

is incomprehensible. "representatives" should be "representations" ? "the modulo 6 of 2n" ?? The heuristic has a prod_{p|n}(p-1)/(p-2), so it is clear that the divisibility by 3 should have an impact. The part "there are tight upper and lower bounds" is stupid: of course there are such bounds and in fact G(n) is one of them, a very precise one if you ask me. So please delete that.

The previous part is well done, except for the last heuristic: the condition p_1 <= p_2 \le ... \le p_c is unusual in this part of the theory and is used only in the algorithmic part sometimes. As a consequence the heuristic in case c = 2 is slightly difficult to understand and leads to an error: the integral should be from 2 to n/2, so that the complete heuristic should be divided by 2. The heuristic given is the classical one where the condition $p_1\le p_2$ is omitted. This error has been there for a long time, but I was lazy and did not mention it up to now. Best is to remove the conditions p_1 <= p_2 \le ... \le p_c and, in the integral: x_1 <= x_2 \le ... \le x_c.

As for credentials, please ask Olivier Ramaré [myself] or Hugh Montgomery, they'll confirm. }} Eramar (talk) 09:19, 14 January 2023 (UTC)

I have removed the crap, and rewritten the paragraph on Goldbach's comet. The last edit request is more technical, and you should better propose a correct formulation. D.Lazard (talk) 11:48, 14 January 2023 (UTC)

Asymptotic version of the conjecture

I have been privy to a paper claiming to have addressed the "asymptotic" version of both Lemoine and the Binary Goldbach conjecture. 10.14293/111.000/000052.v1 I don't know if the paper has been reviewed and published in a reputable journal. The ideas seems to me to be original, while I cannot decide if it holds any credence.

(Sadinova (talk) 22:55, 3 April 2023 (UTC))

Hi Prof. Agama, I think this sort of unseemly self-promotion is below you. Please also review David Eppstein's comment here. --JBL (talk) 00:04, 4 April 2023 (UTC)

Dear JBL,

It may be below me but it an attempt not for my sake. This might be an intellectual discourse that may serve each one of us.

(Sadinova (talk) 09:58, 4 April 2023 (UTC))

Dear JBL,

I suppose the kind of "unseemly self promotion" you described is acceptable on talk pages but not for editing. I would think that explicit disclosure of identity of culprit editors is not encouraged when investigating issues of COI, except in rare circumstances. Going forward, you may have to address me with my user name and not the name of the true identity of the editor of which you may be inately suspicious.

(Sadinova (talk) 11:08, 4 April 2023 (UTC))

COI editors are allowed under certain circumstances; among them is that they disclose their conflict. - CRGreathouse (t | c) 20:38, 5 April 2023 (UTC)

Algorithms for finding the Goldbach primes for n

I am doing a paper on Goldbach’s Conjecture, and tried finding algorithms that solve the number for n. The closest thing I could find was looping through and looking at p and 2n-p, where p is a prime less than n. Are there any algorithms other than this? 98.97.11.203 (talk) 21:55, 16 May 2023 (UTC)

Yes. Any number n=2m where m is not divisible by 3, can only be made with primes congruent to m mod 3. This roughly halves the search space. Unfortunately I don't have a citation or it would have made it in before now. Roderick MacPhee (talk) 22:37, 27 May 2023 (UTC)

in fact once you eliminate (p,2n-p) you can ignore all primes congruent to 2n modulo p. Roderick MacPhee (talk) 22:41, 27 May 2023 (UTC)

You also can ignore divisors of m Roderick MacPhee (talk) 18:45, 28 May 2023 (UTC)

Visual representation of conjecture

In Goldbach's conjecture you have p+q=2m implies m-p=q-m for equidistance, which can be thought of as a square on the plane. Could we add this for visual learners? Roderick MacPhee (talk) 00:05, 28 May 2023 (UTC)

Not without a reliable source. --JBL (talk) 17:42, 28 May 2023 (UTC)

Some days I feel like I should write a paper... look at my user page. But I'd likely not get through review. For example, the mod argument for search space under algorithms, is simply using the fact that 2n-q would be divisible by p if both are congruent mod p but the only multiple of a prime that is prime is a multiplier of 1. Roderick MacPhee (talk) 17:55, 28 May 2023 (UTC)

Go and write your paper and at least try to get it through review!

The article already has this visualisation: .

Does that come close to your 'square on the plane'? E.g. the visual 'square' for 11+11=22 has three solutions:

• 11+11,
• 17+5 and
• 19+3. Uwappa (talk) 18:11, 28 May 2023 (UTC)

Take (n,n) on the line y=x and make the square with diagonal along 2n-x Goldbach's conjecture has either top left or bottom right corner at (p,q) Roderick MacPhee (talk) 18:16, 28 May 2023 (UTC)

I did look at your user page but did not see any diagram there. Could you create a graph to visualise what you mean?

I think the diagram above is already very close to your suggestion:

1. imagine the diagram rotated, so 0,0 would be in the left hand bottom corner.
2. imagine the red and blue lines to be in a 90 degrees angle. The blue numbers would be on the x axis. Blue lines would be vertical. Red numbers would on the y axis. Red lines would be horizontal.
3. the current numbers 4, 6, 8, ... would be on a diagonal (2,2), (3,3), (4,4), ... (0,0) and (11,11) would make two points of a square.
4. the grey lines would be orthogonal to the diagonal from step 3 and connect points like (11,11) (17,5) and (19,3).

Yes, it will be easy to visualise with squares that m-p=q-m, but that does not help in any way to prove that each grey line runs through at least one point with prime numbers as coordinates.

So I think the diagram that you are envisioning is actually already in the article, albeit in a slightly different design. Visual learners will probably have no problem executing the steps above in their mind. Text thinkers will probably just skip the diagram and focus on text. Uwappa (talk) 19:37, 28 May 2023 (UTC)

I'll have to upload a picture, drawing one with picture or tikzpicture fails Roderick MacPhee (talk) 20:20, 28 May 2023 (UTC)

or I'll let it be, you have a point. Roderick MacPhee (talk) 20:22, 28 May 2023 (UTC)

For any average of two n that have distance d equal to the respective p,q will produce two pairs equal to their sum. This in theory can half the search space again... Roderick MacPhee (talk) 20:27, 28 May 2023 (UTC)

Thank you. I enjoy your visual way of thinking. A search can not prove the conjecture right. A smaller search space would only help if it would prove the conjecture wrong faster. No search has proven it wrong yet.

An extended version to 50 of the diagram:

The visual version of Goldbach conjecture: Each grey line has at least one point in which it crosses both red and blue.

For the grey line at 50 you can see your p and q search space reduction would visually translate to:

• 25 <= red line <= 48
• blue line <= 25

Uwappa (talk) 21:24, 28 May 2023 (UTC)

True, however you might be able to induce it a bit (only if no 2 values n is the average of, have equidistant primes, will n fail to have a pair of n's own) . It's similar to an extension of Green-Tao theorem to composites at last check. Roderick MacPhee (talk) 21:32, 28 May 2023 (UTC)

example we don't need to check 34 because (5,17) and (17,29) are equidistant pairs for 11 and 23 so (5,29) and(17,17) are pairs that sum to 34 Roderick MacPhee (talk) 21:37, 28 May 2023 (UTC)

I am not sure what your goal is. Are you still searching a for a diagram to explain Goldbach's conjecture to visual learners? I think the current diagram does that job well enough. Do you wish to expand the article with more info about used search algorithms? Please do!

Are you trying to prove the conjecture wrong or right by inventing an efficient search algorithm? Well, good luck. Please realise that Wikipedia does not allow original research. Still, just go for it! Write your paper and come back once reliable secondary sources confirm your results. Success! Uwappa (talk) 22:50, 28 May 2023 (UTC)

Yeah, was mostly looking for a diagram. But yeah, I could create a section on search algorithms. In theory, arguments with the modular prime counting function, would help pigeonhole the conjecture in theory. I'm no Mathematician though. Roderick MacPhee (talk) 23:07, 28 May 2023 (UTC)

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, 22 April 2003 (UTC)

Marco,

Chen's result is not identical to Goldbach's conjecture, not even for every number> some unknown number n.— Preceding unsigned comment added by TeunSpaans (talk • contribs) 05:38, 23 April 2003 (UTC)

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, 6 March 2004 (UTC)

Heuristic for primes adding to 2n₃

Only if no solution exist to:

$q_{2}-n_{2}=n_{2}-p_{2}=q_{1}-n_{1}=n_{1}-p_{1}=d$

where

$n_{1}+n_{2}=2n_{3}$ will no solution for $2n_{3}$ exist. Roderick MacPhee (talk) 16:06, 1 June 2023 (UTC)

Inconsistency

The article is not totally consistent about Helfgott's supposed proof. 2A00:23CC:B59B:6B01:E914:3ED4:E410:C8A5 (talk) 15:38, 9 September 2023 (UTC)

I have rewritten the paragraph for clarifying the history of this paper. This history suggests that the first proof was incomplete, that the referee was unable to check the details of the proof, and thus asked for a revision without rejecting the paper. The number of revisions suggests that the demand of the referee revealed to Helfgott a gap in the proof that he is unable to fill. D.Lazard (talk) 16:37, 9 September 2023 (UTC)