Today at my number theory lecture the professor want to prove that the statement below is false.
Every prime number \(p\) can be written in the form \(p = ax+b\) were \(a, b\) are coprime integers, \(a > 1\) and \(x \in \mathbb N\).
She said that we need to find one case where the statement doesn't doesn't hold to be true.
The case she showed was \(4\cdot3+3=15 \) and since 15 is a composite then the statement is false.
I tried to explain to her that this proof is wrong but I couldn't.
So i want to mathematically explain why this proof is wrong, Any help with that?