My main interest is physics. However, being self-taught in mathematics for the most part, my proofs tend to be more intuitive than it is acceptable. Yet, I recognize my inaptitude in rigor, and I would like to proceed in practicing my writing proofs in a more mathematical way. Having no formal instruction, and due to my lack of experience in writing rigorous proofs, the mistake that I make is primarily attacking a problem directly, without being able to formulate a sketch, which implies a, in principle, sloppy result.
I want, if possible, to start generalizing each problem in translating a given theorem in set-theoretical language, and form a sketch of proof in providing the proof for the generalization of the theorem. What are your experiences?
Example of an elementary result:
Theorem. If $a$ and $p$ are natural numbers such that $a^p-1$ is prime, then $a=2$ or $p=1$.
How would that be described in formal, set-theoretical language, and proved from there?
(I understand that the proof is itself trivial, but bear with me.)