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.
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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.