I really think an example like yours (except including the addends), or some other easier to see but valid example that adds to a prime like the first example would be more illustrative.
My example without the addends is sort of the point, right? I don't know what the addends are, but I am absolutely certain they exist. There's a proof. Pick any huge odd number you like, and the same guarantee exists. I'm not mathematician enough to guess how difficult it might be to find said addends, and digging around on Wolfram Alpha long enough to find out sounds too much like work for this time of night. But maybe it's difficult enough to be useful. And maybe not. Encryption is generally built on the difficulties of prime factorization. I don't know how difficult it is to find a triple prime partition of a large odd number, and maybe there's a reason cryptographers prefer factorization to partitioning. Maybe it's difficult enough?