Talk:Goldbach's conjecture

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[edit] Historical claims

ALBERTO DURAN WAS THE ONE WHO SOLVE THIS (FROM CARACAS, VENEZUELA) —Preceding unsigned comment added by 190.6.0.188 (talk) 04:14, 5 December 2007 (UTC)

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)

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)

[edit] 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))

[edit] 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)

[edit] 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)

[edit] Meaning of 'sufficently large'

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

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

Marco,
Chens result is not identical to Goldbachs conjecture, not even for every number> some unknown number n. -- Anonymous
Mkrohn accurately formulates what mathematicians mean by the phrase "sufficiently large". I'll add a link to sufficiently large to make this clear for everyone.
—Herbee 03:43, 2004 Mar 6 (UTC)

[edit] 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)

You are playing unfairly with above, as 1 is a prime according to many (since divisible only by itself and 1). Try to find a more honest counter-argument, which is not based on a technicality. —Preceding unsigned comment added by 81.0.68.145 (talk) 20:58, 15 September 2007 (UTC)

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

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

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

I have moved this into Lawson's conjecture.

[edit] 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)

[edit] 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)

[edit] Goldbach equivelent to Lawson

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

[edit] "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, ...., Pmax are all the primes smaller than N. We can choose N = Pmax+1. Then, at least one of Qk = 2Pmax+2–Pk will be a prime, if Goldbach's conjecture is true. The choices of Pk 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)

[edit] 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.

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)

[edit] 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)

[edit] 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)

[edit] At last a proof?

Can someone look at this: http://arxiv.org/ftp/math/papers/0701/0701188.pdf 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)