If the sum of three primes is equal to a prime ... Does anyone know how to always get a prime from the sum of three primes? 
For example: $5+7+11=23$, $17+29+43=89$, etc.

It has long been conjectured that every odd number greater than $7$ is the sum of $3$ primes. It is known that every large enough odd number is the sum of three primes.  I do not know of any extra information known if the target odd number is itself prime.
If your question has to do with an efficient algorithm for finding the $3$ primes, I know very little. But it turns out that for large odd $n$, there seems to be a large number of representations of $n$ as a sum of three primes, so an efficiently conducted search works reasonably well. 
