Explain this math statement

Question

Quoting a paragraph from Timothy Grover's The Princeton Companion To Mathematics,

(9) ∀n ∃m (m>n) ∧ (m ∈ P).

In words, this says that for every n we can find some m that is both bigger than n and a prime. If we wish to unpack sentence (9) further, we could replace the part m ∈ P by

(10) ∀a,b ab = m ⇒ ((a = 1) ∨ (b = 1)).

Here in (m ∈ P), Timothy meant m belongs to prime. How is (m ∈ P) be rewritten as sentence (10). As I understand when sentence (10) explained in english, for all a and b, ab=m implies that either a =1, b=1 or both being 1. So a=4, b=1 satisfies sentence (10). ab = m = 1*4 = 4 is not a prime number. This is clearly wrong. What am I missing here.

Answer 1

What you are missing is that $a,b$ is preceded by $\forall$, meaning that for all possible pairs of (presumably positive) integers $a, b$ such that their product equals $m$, either $a = 1$, or $b = 1$. This prevents $m = 4$ because there exists the choice $a = b = 2$ for which $ab = 4$ but neither $a = 1$ nor $b = 1$.

Answer 2

You are missing the fact that it has to hold for any $a$ and $b$ ... so if you just find one example this is not sufficient.

For example, for $m=4$, you have $a=2$ and $b=2$ which is a counterexample, so $(10)$ is not true for $m=4$.