Has Goldbach's Conjecture been proven?

Question

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?

Answer 1

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.