Talk:Goldbach's conjecture

Please review the following proof

Prime number is can expressed to " ab + 1 "

Even number is can expressed to " 2k + 1 + 1"

The sum of two prime number is expressed to

" (ab +1 ) + (ab +1) " = 2ab + 1 + 1

threfore,

Prime number ( 2k + 1 + 1) is equal to the sum of two prime number ( 2ab + 1 + 1). Dreamtexter (talk) 03:05, 11 May 2024 (UTC)

Your grammar is so poor, your proof is unreadable.—Anita5192 (talk) 03:14, 11 May 2024 (UTC)

Not to mention farcically wrong. (2k + 1 + 1) is an even number, not a prime. All this does is show the trivially obvious result that doubling a prime gives an even number. Doubling any integer gives an even number. Meters (talk) 04:13, 11 May 2024 (UTC)

Data Analysis Report

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)

Known to Descartes before Goldbach?

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."[1] 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 ⁠$n>2$⁠, either ⁠$n$⁠ or ⁠$n-2$⁠ 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)