For the prime numbers, I wrote the following definitons:(one of the definition has to be the "general definition of the prime numbers" and the other is based on the theorem of irreducible elements).I have to find 2 formulas which have one free variable and the domain are the integers. Are my formulas correct?
irreducible number(or irreducible element?):
if $ \ge 2$
$ \forall a,b \in \mathbb{N}:p=ab \implies(p=a ,$ or $ p=b) $
prime number:
if $ \ge 2$
$ \forall a,b \in \mathbb{N}:p|ab \implies(p|a ,$ or $ p|b) $ (where $|$ is the notation of divisibility)
How to write in first order logic(so with formalization)the following: "Any prime is irreducible"?
I would say that $p$ is prime iff
$$p \ge 2 \land \forall a,b \in \mathbb{N}:p=ab \implies(a=1 \lor a=p) $$
So with $p$ irreducible number iff
$$ p \ge 2 \land \forall a,b \in \mathbb{N}:p=ab \implies(p=a \lor p=b) $$
You can express the claim that any prime is irreducible like this:
$$\forall p \in \mathbb{N}: ('Prime(p)' \Rightarrow 'Irreducible(p)')$$
, i.e.:
$$\forall p \in \mathbb{N}: (p \ge 2 \land \forall a,b \in \mathbb{N}:p=ab \implies(a=1 \lor a=p)) \Rightarrow ( p \ge 2 \land \forall a,b \in \mathbb{N}:p=ab \implies(p=a \lor p=b) )$$