Consider $\mathbb{Z_+}$ as interpretation of a signature, which contains predicate of equality and predicate $x$ divides $y$ aka $x | y$. How do I express a predicate $\exists n\ x = 2^n$, if I add constant $2$ to my signature?
I think I should use predicate "$x = 1$" and predicate "$x$ - is a prime number", which I already expressed as the following:
$x = 1 \iff \forall z \ (x | z)$
$x$ - is a prime number $\iff \neg(\exists y(\neg(y = 1) \land \neg(y = x) \land y|x))$