Talk:Goldbach's conjecture

Deep analysis with R Markdown exploring possibilities of sum of 2 prime numbers

https://www.kaggle.com/code/marcoagarciaa/goldbach-conjecture-data-analysis-report Garcia m antonio (talk) 01:16, 23 October 2024 (UTC)

This was raised briefly @ Talk:Goldbach's conjecture/Archive 1#Historical claims:

: 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''

Since then, nothing. But I now come across the claim in Paul Hoffman's book The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth (1998), in which he writes:

: "Descartes actually discovered this before Goldbach," said Erdős, "but it is better that the conjecture was named for Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor." (Chapter 1 "Straight from the book", p. 36).

Comments? -- Jack of Oz ^{[pleasantries]} 07:09, 9 January 2025 (UTC)

:Descartes wrote that "Every even number can be expressed as the sum of at most three primes."[https://real.mtak.hu/164172/1/PJ_DESCARTES_Conjecture1109.pdf] This is regarded as being equivalent to Goldbach's conjecture, although it is worded differently.--♦IanMacM♦ ^{(talk to me)} 08:44, 9 January 2025 (UTC)

:: Then shouldn't we acknowledge his primacy in the article? -- Jack of Oz ^{[pleasantries]} 10:04, 9 January 2025 (UTC)

:::It could be seen as an example of Stigler's law of eponymy. The conjecture by Descartes was apparently not a well known part of his work, so it was an idea that occurred independently to Goldbach. The article could mention this.--♦IanMacM♦ ^{(talk to me)} 10:12, 9 January 2025 (UTC)

::I don't know what you mean by "regarded as being equivalent", but I don't believe they easily shown to be equivalent. Consider, e.g. that if there was a single exception n to Goldbach's conjecture, then n-2 is not an exception and therefore the sum of two primes, n-2 = p_1 + p_2, and thereby n = p_1 + p_2 + 2 is the sum of three primes and so Descarte's version of the conjecture could still be true (see also https://math.stackexchange.com/questions/4934231/every-even-number-is-the-sum-of-at-most-three-primes).

::The cited source only gives the weaker statement "It is also obvious that if an even N satisfies Descartes Conjecture then N or N − 2 can be expressed as the sum of two primes. The converse is

::clearly also true." SimFis (talk) 21:24, 19 March 2025 (UTC)

:::When adding this to the article, I tried to keep the wording simple. Paul Erdős considered that the wording of Descartes' conjecture also implied Goldbach's conjecture involving the sum of two primes (provided, of course, that an even N satisfies Descartes' conjecture). Whether this is exactly the same as Goldbach's strong conjecture could be debated, although there is a clear similarity. The TL;DR is that Descartes' version could be true even if Goldbach's was not. Does anyone have suggestions for other ways of wording this?--♦IanMacM♦ ^{(talk to me)} 20:07, 24 March 2025 (UTC)

::::Descartes' conjecture can be restated as "given an even integer {{tmath|n>2}}, either {{tmath|n}} or {{tmath|n-2}} is the sum of two primes". So, if Goldbach's conjecture is true, Decartes' conjecture is also true, but the converse is seems as difficult to proof as Goldbach's conjecture. For example, if one could prove that the difference between two counterexamples of Goldbach's conjecture is at least 4, then one would remain far to have a proof of Goldbach's conjecture, but one would have a proof of Descartes' conjecture.

::::So, I have fixed the formulation in the article D.Lazard (talk) 22:20, 24 March 2025 (UTC)

:::::In fact, Descartes' conjecture can be restated as "the difference is at least four between two counterexamples (if any) of Goldbach's conjecture", D.Lazard (talk) 22:32, 24 March 2025 (UTC)

The new section {{alink|Equidistant prime pairs and Goldbach's conjecture}} is very confusing, since it starts with a non-definition: for the given definition, every pair of odd primes is an equidistant prime pair (equidistant from their arithmetic mean). I guess that the intended definition is that of a prime pair equidistant from {{tmath|n}}.

Moreover, a prime pair equidistant from {{tmath|n}} is exactly the same thing as a Goldbach partition of {{tmath|2n}}. As "Goldbach partition" is much more common than "equidistant prime pair", the new section must be merged with {{alink|Goldbach partition function}} for forming a single section to be called {{alink|Goldbach partitions}}. Also, it must be avoided to say that some (7) people worked on the subject without saying what are their result.

Reading the new section more carefully, it appears that it is almost exclusively devoted to Winkelmann's work. This is not acceptaable for a subject on which many authors have worked. So, I'll revert the new section, hoping that somebody will write something encyclopedic on Goldbach partitions. D.Lazard (talk) 18:00, 10 June 2025 (UTC)

:Fair enough, I can see how parts of the proposed section may be confusing or would not satisfy certain requirements. I'm sure we can do better, despite having put some effort into preparing and editing my contribution. I agree that a proper section on Goldbach partitions would make sense. Maybe I can do that, but it'll take some time.

:As for the intended definition of equidistant prime pairs, I see your point in mentioning that, technically speaking, it's the same thing as Goldbach partitions of $2n$ (with {{mvar|p}} and {{mvar|q}} being equidistant from their arithmetic mean). However, I think adding the distance variable would make sense, also including the visualizations of the partitions as described by Winkelmann, which was indeed my primary source and the main reason for adding the new section. The idea of this approach representing a generalized form of twin primes is also intriguing, at least as far as I'm concerned (or twin primes being a "specific case" of equidistant prime pairs, where the distance $d\; =\; 1$ so that $p\_\{1\}=n-d$ and $p\_\{2\}=n+d$).

:The intention was definitely not to exclude or omit other contributions. As mentioned above, a more comprehensive section on Goldbach partitions is probably something we should add in future. If you have any further suggestions or objections regarding such an addition, please do let me know. KalGari81 (talk) 21:21, 16 June 2025 (UTC)

Hi,

the term "strong Goldbach conjecture" is first used early in the page, under "Partial results", but is not defined until the middle of the page, at the end of "Formal statement", and so the distinction made at the initial use is ambiguous. Normally I would make a change of this type myself, but I don't feel comfortable enough with this content to make this change. Can someone who understands this well adjust as appropriate?

Thanks,

Lew Outside2017 (talk) 15:42, 19 June 2025 (UTC)

:Goldbach's weak conjecture is an offshoot that is similar to, but weaker than, the problem that Goldbach and Euler discussed in 1742. When the term "Goldbach's conjecture" is used in a general context, it should be taken as meaning that all even natural numbers greater than two are the sum of two primes. It isn't really necessary to say "strong Goldbach" all the time, but I have adjusted the article wording in an attempt to make this a bit clearer.--♦IanMacM♦ ^{(talk to me)} 17:33, 19 June 2025 (UTC)