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I am a high school student currently self-studying number theory using the book Elementary Number Theory by David Burton. Currently, I'm doing Chapter 3: Primes and their Distribution. I've noticed that often time the questions which come up are in this form-
If [condition], then prove that $p$ is prime.
This is a very general question, but in essence what I'm asking is what condition do you prove a certain number to fulfil so you can say that it's prime? For example, in the case of divisibility, we often use the argument that $a | 1$ to prove that $a=1$ and other general tricks like that.
For instance, in this question-
If $p$ and $p^2+8$ are both prime, then prove that $p^3+4$ is also prime.
Here I'm unable to even begin proving the question not because I don't know what to do, but because I'm unaware what kind of argument I need to use to prove a number is prime.
Is there some general argument that we often apply if we want to prove that a number is prime?