Talk:Goldbach's conjecture

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"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)

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)

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)

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)

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)

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.

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)

Contents

[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] 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

[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)

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