For background I am currently taking Introduction to Abstract Math at college, having taken Calculus 1 and (part of) 2 already.
How would I rewrite the following specific definition of irreducible numbers using quantifiers?
An integer $n > 1$ is irreducible if it cannot be factored as $n = ab$, where $a, b \in \mathbb{Z}$ with $1 < a,b < n$.
EDIT: I forgot to add my original attempt.
An integer $n > 1$ is irreducible if it has the following property: $$\forall a \in \{a \in \mathbb{Z}^+ \mid 1 < a < n\}, \forall b \in \{b \in \mathbb{Z}^+ \mid 1 < b < n\}, (ab \nmid n)$$ It still doesn't look quite right though, and I'm pretty sure I messed it up somewhere.
This works: An integer $n$ is irreducible if $\not \exists a, b \in \mathbb{Z}$ such that $1 < a, b < n$ and $n = ab$. In words, this says there do not exist integers $a$ and $b$ such that $a$ and $b$ are both strictly between $1$ and $n$, and for which $n = ab$.
You could also say $\forall a \in \{a \in \mathbb{Z}^+ \mid 1 < a < n\}, \forall b \in \{b \in \mathbb{Z}^+ \mid 1 < b < n\}, ab \neq n$: for all pairs $a$ and $b$ strictly between $1$ and $n$, their product is not equal to $n$. In your attempt, I don't see how you are defining $p$.