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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?

Binh Ho
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    Welcome to Mathematics Stack Exchange. This Wikipedia article summarizes weaker results that have been proven – J. W. Tanner Aug 19 '19 at 00:53
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    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." https://artofproblemsolving.com/wiki/index.php/Goldbach_Conjecture – NoChance Aug 19 '19 at 00:53
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    Math. Comp. 83 (2014) 2033-2060 reports it's true up to 4⋅10^18 – J. W. Tanner Aug 19 '19 at 01:04
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    https://en.wikipedia.org/wiki/Goldbach%27s_conjecture "Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory" – reuns Aug 19 '19 at 01:25
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    No reward announced for solving it? – Narasimham Aug 19 '19 at 01:49
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    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. – Peter Jan 10 '20 at 09:51
  • Related. – Kurt G. Sep 07 '23 at 06:30

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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.

Daniele Tampieri
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JoshuaZ
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  • 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 –  Aug 19 '19 at 12:02
  • 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 –  Aug 19 '19 at 12:04
  • 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 –  Aug 19 '19 at 12:05
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    @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. – JoshuaZ Aug 19 '19 at 13:45
  • My point was to show your point about cranks ( like myself) –  Aug 19 '19 at 15:00
  • this is an excellent answer for illustrating the problem of verifying claims for validity, (and by extension verifying claims of expertise) in the knowledge domain of mathematics. Do you know of any research done on this general phenomenon? How does one go about "debugging" claims of expertise in all knowledge domains, not just mathematics? – dreftymac Oct 31 '22 at 18:42
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    @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. – JoshuaZ Oct 31 '22 at 18:45
  • Your points re: epistemology and implications and (non)existence of an unambiguous answer are well taken, but perhaps you do not give yourself enough credit. Already you've identified some good heuristic guidelines. For example: 1) is the problem statement accessible to non-experts; 2) what is the volume and variety of references out there; 3) what (if any) is the relative ranking of those references ... (so on) ... I find this phenomenon fascinating, and your answer on this site is the first I've seen that seems to spell it out so succinctly and clearly. So, thanks anyway for your reply! – dreftymac Oct 31 '22 at 19:10