I am not sure whether this is the right forum for this question, so it might be migrated somewhere else; but, it is, I think, certainly germane to the wider idea of pursuing interesting questions and developing their answers.
What are the basics of mathematical research? How do you develop a facility for it, outside of the traditional edifice of the modern university and its relata (e.g. affiliated research groups/corporations)?
(NB: I am implicitly suggesting that these institutions, in general, have a certain internal/insular model of research that does not have anything to doing research, except perhaps that it forms the de facto vanguard of arbitrating mathematical truth, and thus the worthiness of research--there is a post here that suggest that submissions by known amateur is subject to greater scrutiny, and I confess myself puzzled, since the question is one about whether the results presented, assuming they are appropriate to the venue, are sound, not whether one is part of the club, unless he was voucher for by some club member cf. Ramanujan)
In particular, graduate schools often evaluate a Ph.D candidate not so much on their mathematical accomplishments but as much as on whether they have the chops to do research, to find, pursue and interrogate new areas of mathematics, or extend, simplify or interrelate others. Yet it seems, that for all the formality of graduate school, it can certainly be done by the interested and persistent everyman. Certainly, number theory is a field that generates a lot of "amateur" interest, but as the technical prerequisites mount, the involvement of the "amateur" is frowned upon, since "clearly" he will probably just not know what the dedicated "professional" mathematician knows. So then, we might revise my question as follows:
What are the necessary and sufficient conditions of good research in mathematics, not framed under the assumption that one has particular training, but rather in the general sense that one is interested in saying something new, interesting or perhaps simply offering a new perspective?