Prove that L={$a^p$: p is prime } is not regular using pumping lemma

Question

https://cs.stackexchange.com/questions/145675/understanding-about-pumping-lemma-for-regular-language-confusions-of-beginner

This the reference idea I have used in this proof. But I am very clear that my proof is indeed wrong. But I want to learn if there is any way to make this proof right this way? And how am I supposed to do new questions in exam if there is not a working way to solve all proofs? I mean math should be like that no? Are there types of questions that I need to practice in order to be able to solve as many as many problems?

Anyway, here is my proof-:

1. Let $a^p$ is regular

w=$a^n$=xyz

Let y= $a^m$ where m is between 1 to n Now let i=2.

|$xy^2z$|=|xyz|+|y|=n+m

Here

n+1<n+m<n+n

or n+1<n+m<2n. Is there way to prove that n+m isn't prime based on this equation?

Answer 1

No, there is no way to prove such a thing. You can't limit yourself to a single $i$. But it is possible to prove that for $i=1,2,3,\ldots$, the different $n+im$ can't all be prime. In particular, if you pick a prime $p$ that doesn't divide $m$, then $n+im$ is occasionally divisible by $p$.

As for the more general question, there is no magic bullet for easily learning to solve problems you haven't seen before. Practice. Develop a large toolbox of tricks. Develop a gut feeling for when each trick usually works.