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This is a little vague question, but I think this is the best place to ask it.

We haven't found an even number which cannot be written as a sum of two primes, but mathematicians must have studied that what sort of properties such a number should have.

I am asking about those properties.

For example if $n$ is an even number which cannot be written as the sum of $2$ primes then

$n\neq 2p$ for $p$ prime

Because if $n=2p$

$n=p+p$ therefore it can be written as the sum of $2$ primes which contradicts the definition of $n$

So $n$ can't be twice a prime.

Are there any properties like these which mathematicians have found?

Rounak Sarkar
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  • Sure, there are obvious properties of positive integers $n$, which would violate Goldbach's conjecture, namely $n=p+q$ with two primes, the same way you argued for $n=p+p$. But this is just a tautology. – Dietrich Burde Jul 02 '21 at 08:40
  • @Dietrich Burde, I am not a mathematician, so I can only find out obvious facts, that is why I included that simple property in my example. – Rounak Sarkar Jul 02 '21 at 08:43
  • But this doesn't help, because we run in circles. Counterexamples to Goldbach are then counterexamples to $n=p+q$ - yes, that's nothing new. We exactly want to know about the properties of the sum of two primes $p$ and $q$. – Dietrich Burde Jul 02 '21 at 08:44
  • We can only say that $n$ cannot have , lets say , the form $11+p$ with a prime $p$ where "$11$" can be replaced by any prime. If we would have strong necessary conditions for a counterexample that were easy to check, the current search limit would probably be much larger than it is. – Peter Jul 02 '21 at 09:15
  • Moreover, no mathematician seriously believes in the existence of a counterexample , so even if we could establish such conditions, this would not strengthen the conjecture significantly. – Peter Jul 02 '21 at 09:17
  • This question is closed so I can't answer, but check out OEIS A352587 – Goldbug Mar 24 '22 at 20:05

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We can ask ourselfs where not to look for a counterexample to Goldbach. However, there are not too many helpful elementary properties known. The case $n=2p$ has been mentioned already. Another case is that even numbers with a higher than $0.5$ density of primes among the totients are guaranteed to have a Goldbach partition. So if there is a counterexample to strong Goldbach, it is in the region where this density is below $0.5$.

Dietrich Burde
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