a few details might help...I want to formally prove some selected theorems about number theory. How is such a proof constructed and what has to be shown?
Your response is technically true but utterly useless. Would you suggest approaching the abc conjecture or the Goldbach conjecture or (before Wiles) FLT? Do you think that if Wiles saw your advice in the late 1980s when he started work on FLT, he would have slapped his forehead and gone, "Duhhhh, why didn't I think of that!"?A statement written in a formal language (of â€˜number theoryâ€™) has a proof if it is true given the set of axioms (true statements) one chooses to accept as given to be true.
For example, suppose we have an axiom that says 2 is a purple number. Further suppose we have a second axiom that says if $x$ is a purple number, $\exists y = 3x$ such that $y$ is also a purple number. Then we can prove that 18 is a purple number because 6 = 3*2 is a purple number so 18 = 3*6 mist also be a purple number.
Generally, in any formal system sufficient to express the natural numbers (arithmetic), there are true statements that cannot be proven within the system. More specifically, see GÃ¶delâ€˜s incompleteness theorem.