In order to have sum of $2$-primes to be a prime one of the primes must be the prime $2$. However the "distance" between adjacent primes increases as we search along the natural numbers.
For example The number of primes in the range: $10^5:10^5+100$ is $6$
The number of primes in the range: $10^7:10^7+100$ is $2$ $(10000019, 10000079)$
Is it possible we have finite numbers of this type?