Translating to logical notation

Question

How would you change "p is a prime." and "x is an odd number" into a logical notation without defining any new predicates where the domain is all natural numbers. Are we supposed to use P (x)?

Answer 1

$$\forall d(d=1\lor d=p\lor \neg\exists n(n\cdot d=p))\\\exists k(x=2\cdot k+1)$$

The last one is if for if you include $0$ in the natural numbers; if you don't, it should be $2k-1$ instead.