Thanks to @AsafKaragila, I found the Jech's book very useful. Here is the result in page 111 concerning to my question: (Note that in what follows $C_n$ is what I have called $AC(n)$.) I don't know how to put this as a comment. So I am writing this as a new answer.
(S) $\qquad$ There is no decomposition of $n$ in a sum of primes,
$$n = p_1 + \ldots + p_s,$$
$\qquad\quad\;\:$ such that $p_i > m$ for all $i = 1, \dots, s$.
(In particular, if $n > m$, then $n$ is not a prime.)
Theorem 7.15. If $m, n$ satisfy condition (S) and if $C_k$ holds for every $k \leq m$, then $C_n$ holds.
Theorem 7.16. If $m, n$ do not satisfy condition (S), then there is a model of set theory in which $C_k$ holds for every $k \leq m$ but $C_n$ fails.
(Images: Theorem 7.15, Theorem 7.16)