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?
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 $0.5$.