Are there any counterexamples to a generalized Goldbach conjecture?

The Goldbach conjecture suggests that every even integer greater than $2$ can be expressed as a sum of two primes. For example:

$$10 = 5+5$$ $$12 = 7+5$$ $$14 = 7+7$$

What is so special about even integers? Can we generalize the conjecture as follows?

Every integer multiple of $n$ greater than $n$ can be expressed as a sum of $n$ primes.

For example:

$$15 = 3+5+7$$ $$21 = 7+7+7$$ $$27 = 7+7+13$$

In the above example, I didn't include $24$, because I'm adding the additional restriction that the number is not a multiple of a prime less than $n$.

• My edit was to replace 27=7+9+11 with 7+7+13 because 9 is not prime. – DanielWainfleet Apr 6 '18 at 18:33
• Oops. >red face< – Adam Hrankowski Apr 6 '18 at 18:50

Fix your whole number $n \ge 3$. Clearly $2n$ is a sum of $n$ primes: $$2n = \underbrace{2 + 2 + \cdots + 2}_\text{n times}$$
If $k \ge 3$ you can write $$kn = \underbrace{2 + 2 + \cdots + 2}_{\text{n-2 times}} + 2m$$ if $kn$ is even, and $$kn = \underbrace{2 + 2 + \cdots + 2}_{\text{n-3 times}} + 3 + 2m$$ if $kn$ is odd. According to Goldbach, in each case $2m$ can be written as a sum of two primes so that $kn$ is a sum of $n$ primes.
• This doesn't seem to take into account the added restriction that $n$ is the lowest prime factor. Or am I missing something? – Adam Hrankowski Apr 6 '18 at 18:16
• To the proposer: In the A it doesn't seem to matter what the prime factors of $kn$ are. For example, with $n=3$ we have $24=2+11+11.$ – DanielWainfleet Apr 6 '18 at 18:47