Write a formula with $\mathcal L_{NT}:=\{0,S,+,\cdot,<,E\}$ for "$x$ is prime" 
Write a formula with the language $\mathcal L_{NT}:=\{0,S,+,\cdot,<,E\}$ for "$x$ is prime".

Here $E$ is the exponentiation function, that is $E(a,b)=a^b$, and $S$ is the successor function. Then assuming that $P(x)$ is the assertion that "$x$ is prime" I need to write a formula for this statement using the above language, but I dont have a clue about how I can write it without using the concept of the set of natural numbers.
This exercise was taken from the book A friendly introduction to mathematical logic of Leary and Kristiansen. Some help will be appreciated, thank you!
 A: I don't know the full details of the question, but it looks possible that there's no need to worry about restricting quantifiers or talking about the set of natural numbers, since for the purposes of the question the natural numbers are the only objects that there are. 
If you wanted to guarantee that all your objects are natural numbers, you would have to say that every number is either zero or the result of applying the successor operator to $0$ finitely many times. For this, you would need a formula along the lines of the axiom of induction, which you can't provide in first-order logic.
If there were objects under consideration which aren't natural numbers, you can't restrict your quantifiers to natural numbers (to say "x is prime if there exists no natural numbers such that ...") without using second-order logic.
This means I suspect that the question is a simple one, and worrying about the domain of quantification isn't part of it. In this case you could simply use a formula such as:
$$\neg \exists y \exists z ( \neg y = x \wedge \neg y = S(0)  \wedge y \cdot z = x ) $$
