Talk:Goldbach's weak conjecture

Early text

Firstly, I believe you mean "add three to any even number > 4 to get the odd numbers > 7". Secondly, the term upper bound used here is confusing. I first read it as the largest number for which the property is known. I see that it is the upper bound for the threshold that determines whether a number is large -- Hari

Good points. Do you want to make the change? AxelBoldt

What if one of the two primes in the strong Goldbach conjecture is three? Then we can't simply apply the formula above, as no prime can be repeated in the sum. -- The Anome

In the weak Goldbach conjecture, you are allowed to repeat primes in the sum. AxelBoldt

Thanks for the clarification -- The Anome

Simplifying

I propose to simplify this sentence:

K. Borodzin proved that 3^14,348,907 is an upper bound for the threshold that determines if a number is large.

to:

K. Borodzin proved that 3^14,348,907 is large enough.

It is fully understandable within context. With this sentence everybody will understand that it is pointless to use larger number (because 3^14,348,907 is large enough) as a threshold that determines if a number is large - so it is natural that this number is an upper bound for the threshold.

and this sentence:

In 2002 Liu Ming-Chit and Wang Tian-Ze lowered this upper bound to approximately 10¹³⁴⁶.

to:

In 2002 Liu Ming-Chit and Wang Tian-Ze lowered this treshold to approximately 10¹³⁴⁶.

CoperNick 10:44, 7 May 2007 (UTC)

I agree. Remember h in threshold if you make the edit. PrimeHunter 00:07, 8 May 2007 (UTC)

Shouldn't the conjecture be every odd number greater than 5? —Preceding unsigned comment added by 96.18.21.83 (talk) 04:21, 16 June 2008 (UTC)

No. It correctly says: "Every odd number greater than 7 can be expressed as the sum of three odd primes." The number 1 is not considered a prime so 7 cannot be expressed as the sum of three odd primes. PrimeHunter (talk) 11:41, 16 June 2008 (UTC)

Difference between two results

The article does not distinguish sufficiently between the 1923 and 1997 results. JMK (talk) 05:54, 30 December 2008 (UTC)

Primality testing

How does a sufficiently efficient primality test help to determine whether the conjecture is true for an odd number of similar size, as claimed in the article? I.e. how do you know how many decompositions you must try till one consists of three prime numbers? --Roentgenium111 (talk) 23:57, 15 November 2010 (UTC)

With currently known primality tests it is much faster to find large probable primes (prp's) than to prove whether they are really prime (unless the numbers are of certain special forms). You don't know for certain how many decompositions you must try. However, experimentation and heuristic arguments indicate that you can "always" find a decomposition with prp's relatively quickly. The slow step will be to prove that the prp's are really prime (unless perhaps if you found a special triple where all three are of the forms that are fast to prove, or the largest is of such a form and the others are so small that it doesn't matter). Here is an original research estimate (based on extensive prime search experience [1]) for a random odd 1346-digit number with the best available programs for PC's: Decomposing into a 1346-digit prp and two tiny primes will on average take around 30 GHz seconds. Proving primality of the 1346-digit prp will on average take around 4 GHz hours. PrimeHunter (talk) 01:05, 16 November 2010 (UTC)

Thanks for your calculations, though that doesn't quite answer my question. I know that heuristic arguments indicate that you can always find a decomposition with primes quickly, but the whole point of the Goldbach conjecture (as it stands today) is to prove that this heuristics holds for every single number. So my question is how long it would take to validate a given counterexample of order 10^1346? I would suspect this can not be done fast enough (e.g. in polynomial time), not even probabilistically.--Roentgenium111 (talk) 16:11, 16 November 2010 (UTC)

Unfeasible

"infeasible with current technology" (I think unfeasible is meant) is misleading. This is like saying "it's not feasible to travel faster than the speed of light with current technology". Everything in computational theory suggests that even if the entire Universe were an idealized enormous parallel computer, it could not possibly process that number of calculations prior to its own entropic demise. It's not a question of "feasibility", it is quite literally impossible based on the conditions of the Universe in which we reside, unless you mean "estimated" results via quantum computing and typically for math problems these are no better than a search over small numbers.

I'm going to alter that line now, but I wanted a record here before I do in case of disagreement. TricksterWolf (talk) 03:39, 28 May 2012 (UTC)

This is a math article. Your edit [2] is more about speculative physics and sounds partially like original research. Lots of things about the Universe (like what is truly an elementary particle if that's even a valid way to describe the Universe) may still be undiscovered. And I'm not sure your statement is even consistent with the current understanding when you for example ignore that n particles can be configured in far more than n states. If reliable sources haven't described it like this then it doesn't belong here. I have changed the original "infeasible with current technology" to "completely infeasible". wikt:infeasible is more common than unfeasible. PrimeHunter (talk) 11:36, 28 May 2012 (UTC)

Misplaced statement?

Currently the article contains the following statement:

Olivier Ramaré in 1995 showed that every even number n≥4 is in fact the sum of at most six primes.

How is that relevant to the weak Goldbach conjecture which only deals with representing odd numbers as sums of primes? I propose to move this statement into the article Goldbach's conjecture. -- Toshio Yamaguchi 12:18, 25 January 2013 (UTC)

On the other hand from this it readily follows that every odd number ≥ 5 is the sum of at most seven primes. -- Toshio Yamaguchi 20:41, 25 January 2013 (UTC)

I clarified how this is relevant to the weak conjecture. -- Toshio Yamaguchi 16:24, 26 January 2013 (UTC)

K. Borozdin's result

Is there a reference for the fact that "K. Borozdin proved, in 1956, that 314348907 is large enough."--212.149.249.123 (talk) 00:47, 12 May 2013 (UTC)

I guess the paper is K. G. Borodzkin, "On I. M. Vinogradov's constant", Proc. 3rd All-Union Math. Conf., vol 1, Izdat. Akad. Nauk SSSR, Moscow, 1956. (Russian) MR 20:6973a. But I don't know any Russian so I can't check the proof. --212.149.249.123 (talk) 21:34, 13 May 2013 (UTC)

possible proof upcoming

https://plus.google.com/114134834346472219368/posts/8qpSYNZFbzC

Should we remove the tag "Conjectures about prime numbers"? --212.149.249.123 (talk) 18:17, 14 May 2013 (UTC)

Has that been peer-reviewed and published in a mathematical journal? Or is Helfgott considered a significant enough expert such that this is not needed? If not, and thus the mathematical community still considers it an unsolved problem, then I think the tag should remain. -- Toshio Yamaguchi 18:29, 14 May 2013 (UTC)

I guess that at least Terence Tao has checked the proof. He wrote in Google+ the following: "Busy day in analytic number theory; Harald Helfgott has complemented his previous paper http://arxiv.org/abs/1205.5252 (obtaining minor arc estimates for the odd Goldbach problem) with major arc estimates, thus finally obtaining an unconditional proof of the odd Goldbach conjecture that every odd number greater than five is the sum of three primes. "--212.149.249.123 (talk) 18:35, 14 May 2013 (UTC)

See also the Mathoverflow discussion.—Emil J. 19:45, 15 May 2013 (UTC)

Imho it is still a bit early and afaik there hasn't been a thorough review yet nor does Tao really say that he had checked the proof as such, he is just providing some info about it. I just posted that above for people to take notice and that very soon (but not yet) it might have to integrated into the article and the article name probably changed from conjecture to theorem.--Kmhkmh (talk) 20:02, 14 May 2013 (UTC)

I think this should be removed until either it has been published in a peer-reviewed journal or Tao has officially confirmed the correctness of the proof. -- Toshio Yamaguchi 10:40, 15 May 2013 (UTC)

7?

Why does this article repeatly say "greater than 7" when the conjecture holds for 7 as well (but not for 5)? — Preceding unsigned comment added by 96.248.226.133 (talk) 00:11, 20 May 2013 (UTC)

It doesn't hold for 7. The article correctly says "Every odd number greater than 7 can be expressed as the sum of three odd primes." 7 = 2+2+3, but 2 is not odd. And 7 = 1+3+3, but 1 is not considered a prime. PrimeHunter (talk) 01:20, 20 May 2013 (UTC)

But that's NOT what the article says. It says "Every odd number greater than 5 can be expressed as the sum of three primes", right at the top. It looks as though the statement of the conjecture hase been edited, but the associated changes have not been carried through properly. Almost immediately afterwards it says "(Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7)". But from the statement of the conjecture as already given, that can be replaced with "(Since if every even number greater than 2 is the sum of two primes, merely adding 3 to each even number greater than 2 will produce the odd numbers greater than 5)".

Clearly, the statement without "odd" is weaker (it merely admits extra candidate decompositions, namely those consisting of 2 + 2 + an odd prime). Are there odd numbers of the form 4 + odd prime (other than 7!) for which there is no other decomposition? If so, of course, the stronger conjecture is false. Off-hand, I don't see why there should be any, though - very likely because I'm stupid. Can someone explain why the conjectures are really equivalent, or why the stronger conjecture is false (!!!)? Failing that, we have two different conjectures, and a historical question about what Goldbach had in mind. — Preceding unsigned comment added by ArnoldTheFrog (talk • contribs) 17:23, 2 May 2017 (UTC)

Major and minor arc estimates

These terms need to be defined if you want to use them in a WP article. - Frankie1969 (talk) 20:09, 30 December 2013 (UTC)

I haven't looked at this closely and can't say I understand all the details, but what is meant by arcs seems to be explained in http://arxiv.org/abs/1305.2897 in section 1.3. -- Toshio Yamaguchi 21:42, 30 December 2013 (UTC)

Using undefined terms is not entirely wrong. They can be understood by a specialist, even if not by most. A definition might be too long to include in a short article.

Major arcs and minor arcs redirect to Hardy–Littlewood circle method, where the terms are briefly desscribed. Deltahedron (talk) 17:06, 22 September 2014 (UTC)

"Helfgott's claim covers both versions of the conjecture."

Perhaps it is better to say that he proved the version about odd primes, as the version with any primes follows directly from that (just note as the last sentence of the proof that 7 = 2 + 2 + 3). If he had proven the version about any primes, however, the version about odd primes would not follow from that (as then perhaps we could express an odd number as the sum of three primes in only one way, but two of these primes were 2). Double sharp (talk) 14:22, 31 October 2015 (UTC)

2002 result updated which threshold?

"In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered this threshold to approximately..."

Which threshold is being referenced? It seems this paragraph was once immediately after the first paragraph of the section, which mentions a threshold of $3^{3^{15}}$. If that's the correct reference, perhaps this paragraph should be moved back to immediately after the first paragraph.

Also seems incongruent that the 1997 results are described prior to the 1995 results paragraphs, even if the 1995 results have a 2012 extension.

Se7ens (talk) 06:25, 21 February 2017 (UTC)

Proven?

The first sentence in the lead says

In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that

Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum).

This leaves the impression that it’s still a conjecture, whereas a few paragraphs later it is said to have been proven. If my understanding is correct, I recommend that the lead sentence be altered to

In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, is the now-proven statement that

Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum).

Loraof (talk) 19:28, 26 November 2017 (UTC)

Computer Search

In the article the following claim appears "Computer searches have only reached as far as 10^18 for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture". However, if Helfgott's article on the proof of the weak Goldbach conjecture it is claimed that these have been done up to 10^30 (and in fact were necessary up to 10^27 to finish the proof) for the weak conjecture at least. I would be surprised if this were not also true for the strong conjecture as well.

Perhaps someone with some more expertise in the area could look up the latest results. — Preceding unsigned comment added by 2001:630:E4:42E0:4823:551A:2C42:4268 (talk) 09:35, 18 September 2018 (UTC)

Original Source of Weak Conjecture

As described in the origins section of the Goldbach's Conjecture article, Goldbach stated two equivalent forms of the strong conjecture during his correspondence with Euler. Ignoring the "greater than 5/2/etc." clauses for simplicity, those conjectures were (i) every integer is the sum of three primes and (ii) every even integer is the sum of two primes, and the weak conjecture is (iii) every odd integer is the sum of three primes, or the slightly stronger (iii)' every odd integer is the sum of three odd primes. The origins section of this article equivocated between (ii) (a version of the strong conjecture) and (iii) (the usual version of the weak conjecture), presenting the origins of the former rather than the latter. I fixed that, but I could not actually find any reference describing the origins of (iii). In their 1923 work on the weak conjecture (available here: https://projecteuclid.org/euclid.acta/1485887559), Hardy and Littlewood do not credit it to Goldbach but instead simply credit the strong conjecture to him and present their proof of the weak conjecture given the generalized Riemann Hypothesis as a step towards solving his original conjecture. Were they the first to explicitly formulate the weak version of the conjecture, or did Goldbach discuss it himself? David815 (talk) 03:03, 26 August 2020 (UTC)