Not every sum is of three primes. 15 = 5 + 5 + 5 which are three primes, but 15 = 2 + 3 + 10 which are not. The theorem says there there is such a sum, not

Not every sum is of three primes. 15 = 5 + 5 + 5 which are three primes, but 15 = 2 + 3 + 10 which are not. The theorem says there there is such a sum, not that all sums are of that form.