Misplaced Pages

Revision as of 11:59, 20 May 2014

This is the talk page for discussing improvements to the Goldbach's conjecture article. This is not a forum for general discussion of the article's subject.

Put new text under old text. Click here to start a new topic. New to Misplaced Pages? Welcome! Learn to edit; get help. Assume good faith Be polite and avoid personal attacks Be welcoming to newcomers Seek dispute resolution if needed Article policies Neutral point of view No original research Verifiability

Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL

Archives: 1Auto-archiving period: 31 days

Mathematics C‑class High‑priority

This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics C This article has been rated as C-class on Misplaced Pages's content assessment scale. High This article has been rated as High-priority on the project's priority scale.

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

• 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)

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,

Chen's result is not identical to Goldbach's 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)

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. — Preceding unsigned comment added by 67.5.135.253 (talk • contribs) 17:15, 24 April 2004 (UTC)

Proof: p,q odd primes ; n,l out of N with n>l

p*q = n²-l² = (n+l)(n-l) => p=(n-l) and q=(n+l)

=> (n+l)+(n-l) = 2n = p+q

For every (n+l) prime is (n-l) also prime. — Preceding unsigned comment added by 84.135.3.157 (talk) 13:09, 7 April 2014 (UTC)

You refer to Talk:Goldbach's conjecture/Archive 1#Lawson's Conjecture. It was added to the article by a "Bill Lawson" in 2003 and quickly removed. I guess he named it after himself and I haven't found mention of it elsewhere. If the pair of primes is allowed to be two identical primes then it's trivially equivalent to Goldbach's conjecture as you show. It's a non-notable reformulation and shouldn't be mentioned in the article. PrimeHunter (talk) 13:50, 7 April 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 4×10 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)

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 Misplaced Pages. PrimeHunter (talk) 11:27, 13 May 2014 (UTC)

COMMUTATIVE ALGEBRA AND ALGEBRAIC TOPOLOGY TO PROVE THE GOLDBACH's CONJECTURE

WARNING . My previous posted talks, (May 18 2014 and May 19 2014 - integrally reported below), have been deleted with the following statement signed by D. Lazard:

... you are showing that the sum of two odd primes is an even number. This shows nothing about every even n....

This claim has noting to do with my paper: it is completely false ! It shows that the reader did not carefully read my paper. Since he insists for the second time with this skin-deep comment, despite my more detailed second post, he shows vandalism. In this Misplaced Pages talk-place it cannot be allowed such a behaviour !! In particular, it appears that D. Lazard does not work in Algebraic Topology and in (co)bordism groups. Therefore, probably he did not read (or understood ) the central part of my proof. By the way, since this is a talk he could interact with me by a polite comment and not deleting my post-talk !

In the following I integrally reproduce my previous talk May 19, 2014.

As I have already announced on Misplaced Pages, the Goldbach's conjecture has been completely proved by Agostino Prástaro in the following paper published on arXiv:

A. Prástaro, The landau's problems. I: The Goldbach's conjecture proved.

With this respect, it may be useful to underline that on 13 Aug 2013, I posted the complete version of my proof of the Goldbach's conjecture, (completing a previous version announced on 13 Aug 2012). Since, it appears that some reaction from experts on the Goldbach's conjecture continues to be negative, and completely outside the real content of my paper, I would like with this Misplaced Pages-talk to help at least interested mathematicians that aim seriously understand my research.

'Warning. The algebraic and topologic methods used to proof the Goldbach's conjecture are not standard in the usual Number Theory.

1) - The first method is focused on a new concept of bordism group (Goldbach-bordism groups), hence it is placed in the Algebraic Topology. By means of this mathematical tool it is possible to decide whether for any fixed positive integer n there exists

at least a Goldbach couple in the interval (0,2n] of integers, i.e., two primes p and q such that p+q=2n. This is really the great novelty of my proof. (See Lemma 2.43.)

2) - The second method uses Commutative Algebra, and in particular Noether rings, Artin rings, maximal ideals, ..., to built all possible solutions of the Golbach's conjecture for any fixed positive integer n. These mathematical tools are standard, but the novelty is in the way where they are used in connection with the Goldbach-bordism groups.

It my be understandable the difficulty to accept this my proof of the Goldbach's conjecture, by experts on Number Theory, since the mathematical tools considered there are outside Number Theory. On the other hand it was easy to foresee that such a long standing open conjecture it would require some new methods to be proved !

In the second part of the paper, posted on arXiv with the title The Landau problems. I-II, 42 pages, are solved also the other three landau's problems. See the following paper.

A. Prástaro, The landau's problems. II: Landau's problems solved.

With this respect it is useful to underline that by introducing new bordism groups, it has been possible to prove two Landau's problems, similarly to what made for the Goldbach's conjecture. (2. Twin prime conjecture. 3. Legendre’s conjecture.) Instead for the fourth Landau's problems (4. Are there infinitely many primes p such that p-1 is a perfect square ?) these methods cannot be applied. (This fact should also be useful to better understand the meaning of those new algebraic topological approaches.) In fact the fourth Landau's problem has been solved by using some diophantine equations of Ramanujan-Nagell-Lebesgue type.

In part I some interesting applications of the Goldbach's conjecture are also given. In particular some relations to quantum algebras in the sense of A. Prástaro, and crystallographic groups are stressed too.(Recall that A. Prástaro proved that integral bordism groups of some

PDEs can be interpreted as extended crystallographic groups.) For related information on this subject see the following papers and Prástaro’s works, quoted therein.)

A. Prástaro, (Un)stability and bordism groups in PDE's, Banach J. Math. Anal. 1(1)(2007), 139--147.

A. Prástaro, Extended crystal PDE's stability.I: The general theory, Math. Comput. Modelling 49(9-10)(2009), 1759--1780.

A. Prástaro, Extended crystal PDE's stability.II: The extended crystal MHD-PDE's, Math. Comput. Modelling 49(9-10)(2009), 1781--1801.

A. Prástaro, On the extended crystal PDE's stability.I: The n-d'Alembert extended crystal PDE's, Appl. Math. Comput. 204(1)(2008), 63--69.

A. Prástaro, On the extended crystal PDE's stability.II: Entropy-regular-solutions in MHD-PDE's, Appl. Math. Comput. 204(1)(2008), 82--89.

A. Prástaro, Extended crystal PDE’s, Mathematics Without Boundaries: Surveys in Pure Mathematics. (Eds. P. M. Pardalos and Th. M. Rassias.) Springer-Heidelberg New York, Dordrecht, London (to appear). arXiv:0811.3693.

A. Prástaro, Quantum extended crystal PDE's, Nonlinear Studies 18(3)(2011), 447--485. arXiv: 1105.0166.

A. Prástaro, Quantum extended crystal super PDE’s, Nonlinear Analysis. Real World Appl.13(6)(2012), 2491–-2529. arXiv:0906.1363.

A. Prástaro, Exotic heat PDE's.II. Essays in Mathematics and its Applications. (Dedicated to Stephen Smale.) (Eds. P. M. Pardalos and Th. M. Rassias.) Springer-Heidelberg New York Dordrecht London (2012), 369--419. arXiv: 1009.1176.

A. Prástaro, Exotic n-d'Alembert PDE's and stability. Nonlinear Analysis: Stability, Approximation and Inequalities. (Dedicated to Themistocles M. Rassias for his 60th birthday.) G. Georgiev (USA), P. Pardalos (USA) and H. M. Srivastava (Canada) (eds.), Springer Optimization and its Applications Volume 68(2012), 571--586. arXiv: 1011.0081.

A. Prástaro, Quantum exotic PDE’s, Nonlinear Anal. Real World Appl. 14(2)(2013), 893–-928. arXiv:1106.0862.

It is clear that in order to understand the technicalities it is necessary to have the patient to carefully read Prástaro's works !

Agostino.prastaro (talk) 11:59, 20 May 2014 (UTC)

• C-Class mathematics articles
• High-priority mathematics articles