Questions about composite numbers Consider the following problem:
Prove or disprove that if $n\in \mathbb{N}$, then $n$ is prime iff $$(n-1)!+n$$ is prime.
If $n$ is composite and greater than $1$, then $n$ has a divisor less than $n-1$, therefore $(n-1)!$ and $n$ have a common factor. Thus "$\Leftarrow$" is true. To proof the other direction we can consider the more general problem:
Let $n\in\mathbb{N}$. Consider the set $$C(n)=\{m\in\mathbb{N}:n+m\text{ is composite}\}.$$ How can we characterize the elements of $C(n)$?
The ideal answer would be to describe all elements in $C(n)$ in terms of only $n$. But, is that possible?
As a first approximation to solve this, we can start by defining for $n,p\in\mathbb{N}$: $$A(n,p)= \{ m\in\mathbb{N}:n+m\equiv 0\pmod{p} \}.$$ After of some observations we can prove that
$$A(n,p)=\{(\lceil n/p \rceil + k)p - n:k\in \mathbb{N}\}$$
and then $A(n,p)$ is the range of a function of the form $f_{n,p}:\mathbb{N}\to \mathbb{N}$. From this $$C(n)=\bigcup_{p=2}^\infty A(n,p),$$
But this still far from a characterization in terms of $n$. What do you think that is the best that we can do or the best we can hope?

One reason your professors might have smiled at you is that
$$ C(n) = C(0) - n, $$
where $C(0) = \{m \in \mathbb N: m \text{ is composite} \}$.  So characterizing $C(n)$ reduces to characterizing $C(0)$, which in turn reduces to characterizing the set of primes $\mathbb N \setminus C(0)$.
(Well, okay, technically $C(n) = (C(0) - n) \cap \mathbb N$ as you've defined it, but cutting off the negative part of the set doesn't make any fundamental difference.)
