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When I searched for the proofs for Goldbach's Conjecture, there seems to be a handful (or more) of papers that attempt to solve it. Are there any official proofs out there yet?

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    $\begingroup$ Welcome to Mathematics Stack Exchange. This Wikipedia article summarizes weaker results that have been proven $\endgroup$ Aug 19, 2019 at 0:53
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    $\begingroup$ This site says "The Goldbach Conjecture is a yet unproven conjecture stating that every even integer greater than two is the sum of two prime numbers. The conjecture has been tested up to 400,000,000,000,000." artofproblemsolving.com/wiki/index.php/Goldbach_Conjecture $\endgroup$
    – NoChance
    Aug 19, 2019 at 0:53
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    $\begingroup$ Math. Comp. 83 (2014) 2033-2060 reports it's true up to 4⋅10^18 $\endgroup$ Aug 19, 2019 at 1:04
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    $\begingroup$ en.wikipedia.org/wiki/Goldbach%27s_conjecture "Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory" $\endgroup$
    – reuns
    Aug 19, 2019 at 1:25
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    $\begingroup$ This conjecture appears to be so hard that it would be almost shocking if someone actually would be able to prove it. It is also a good candidate for a case of Goedel's results. It is well possible that it cannot be proven at all. $\endgroup$
    – Peter
    Jan 10, 2020 at 9:51

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No, Goldbach's Conjecture is still open. We know it is true up to very large $n$ (around 4*10^18). We know also that every sufficiently large even number is the sum of a prime and a number with at most two distinct prime factors: this is Chen's Theorem. We have a variety of other results; for example we know that in a certain rigorous sense, exceptions must be rare.

More broadly why are you seeing "papers" claiming to prove Goldbach's conjecture? The problem is one of mathematical cranks, people who often don't know much mathematics and think they have earth-shattering results and proved major problems. These people are very fond of claiming to have completely solved major problems, and they are particularly attracted to problems where the problems are easy to state (like Goldbach's conjecture, or whether there are any odd perfect numbers, etc.) Until Andrew Wiles, a common crank target was Fermat's Last Theorem, and one still sees cranks claiming to have completely elementary proofs of it.

This is a problem since it makes it harder for non-mathematicians to tell what to pay attention to. As a general rule of thumb, if you don't know if a a paper should be paid attention to, one good thing to do is to check if the paper is in a journal listed on MathSciNet. That's a good first step to see if the paper is one one should take at all seriously. This is a very low bar, since some journals, even those indexed by MathSciNet, have poor quality control, but it is a good way to start. In general, there are a lot of claims of this sort out there, and mathematicians generally have better things to do with their time than to identify what and report to everyone what exactly is wrong with each such claimed solution. Another good check is to see what Wikipedia says: if the stable version of a page mentions that the problem is solved, that's a good sign.

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  • $\begingroup$ my half effort attempt: Goldbach's conjecture, is usually stated as the following: For all even numbers $x>6$ , there exist a pair of odd primes $p,q$, such that $x$ is the sum of $p$ and $q$. This can be restated in math notation as:$$\forall x>6 , x\in 2\mathbb{N}\exists p,q \in \mathbb{P}_{>2},\text{s.t. } p+q=x$$ Goldbach, has many necessary conditions, related to it's consistency with other mathematics. What follows, are just some of these. 1/3 $\endgroup$
    – user645636
    Aug 19, 2019 at 12:02
  • $\begingroup$ As Goldbach is about an even sum, It follows we can divide both sides by 2 . Letting $x=2n=n+n$, we get:$${p+q\over 2}= n$$. Or, using the second half of the equality: $$q-n=n-p=d$$ That is, they are equal distance from their arithmetic mean. Common properties, of products of same parity integers, include being a difference of squares. A property of the sum of squares of such arguments, is they are twice another sum of squares.If you believe Goldbach meant to use distinct odd primes, then you believe there are infinitely many primes in arithmetic progressions of length 3. 2/3 $\endgroup$
    – user645636
    Aug 19, 2019 at 12:04
  • $\begingroup$ Finding a Goldbach partition of $2n$ implies that those primes aren't factors of $n$. This happens via the distributive property. Because of equidistance to odd primes, if one is lower than $n$ then the other is higher. This then implies, via a limitation on the lower, that the higher is between n and 2n-2. Via the sieve of sundaram, we encounter that any product of safe primes, is necessarily 1 or 5 mod 8 If you draw two squares, in the negative direction for width, and positive for height from $(n,n)$; 3/3 $\endgroup$
    – user645636
    Aug 19, 2019 at 12:05
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    $\begingroup$ @RoddyMacPhee I don't think that speculation about how one might go about proving Goldbach's conjecture is productive here. There almost certainly is not any simple proof of Goldbach's conjecture. So many people, both professionals, and amateurs have spent a large amount of time thinking about it. If there were a simple proof, it would almost certainly have been discovered long before now. $\endgroup$
    – JoshuaZ
    Aug 19, 2019 at 13:45
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    $\begingroup$ @dreftymac I don't have a good answer to that unfortunately. If anyone had a really good answer that was universal to all domains, my guess is the world would look very different. In general, when one has expertise in a domain, one can tell, but the difficulty of domains that are far from one's own is serious and raises very big epistemological issues. $\endgroup$
    – JoshuaZ
    Oct 31, 2022 at 18:45

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