The odd number sums follow from the fact about the even numbers.

Why though?

If you can make any even number with 2 primes, you can make an odd number by subtracting one odd prime from the odd number first (e.g. 3) and the resulting even number with 2 more primes.

We don't know whether it's true.

Well, then it's a great mystery of up to 10^18: "T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 × 10^18"