Excuse me for the awkward wording. I'm new to logic. What I really mean is this:
Consider the number theory that spawns from the structure $N=\{\mathbb{N},+,\cdot\}$ (equipped with the usual interpretation) using first order logic. I understand that the formal sentences generated this way are capable of expressing all elementary number theory statements in the sense that the statement "$x$ divides $y$" can be expressed as $\exists z(x\cdot z=y)$, so that $N\models \exists z(x\cdot z=y)$ iff $x$ divides $y$.
But there's a kind of statement that I don't know how to express in the system, like "$x$ can be written as the sum of cubes". It is easy to write "$x$ can be written as the sum of $n$ cubes" given specific $n$ into something like $\exists z_1z_2...z_n(x=z_1\cdot z_1\cdot z_1+z_2\cdot z_2\cdot z_2+...+z_n\cdot z_n\cdot z_n=x)$. But in the first statement it seems one can't follow suit because the "$n$" there is not specific. How can we quantify the unspecific parameter? What are tricks for expressing it?
Edit: By the way, "...that the formal sentences generated this way are capable of expressing all elementary number theory statements..." Is this true?