I'm in a bit of a rut. I find myself quite old (20 - old man) and I don't even know high school math properly. And yet I can actually conduct original research (proof: rediscovered a famous theorem in number theory simply by generalizing a totally unseeming olympiad problem and working at it for a few months, until the result popped out as if by magic). However, I spent the past few years solving olympiad problems, so my knowledge stayed at the same level - basic olympiad level (elementary maths and a tiny pinch of calculus). Going to work after high school didn't help. I blame myself, but I'm seeking your help.
Basically, I'm looking to get to the bleeding edge of research in algebraic and analytic number theory and to accumulate the necessary knowledge in the various prerequisite fields of maths. This, in the shortest possible time. Maybe I ask too much of you, but a large chunk of it reduces to finding the right books on abstract algebra and analysis that will help me accumulate vast amounts of knowledge in these fields while starting nearly from scratch. The books need to be challenging (concise) for this to work: I tried reading Apostol's Calculus and found it too straightforward and not "original" enough if that makes sense. I liked Zorich's Analysis much more, which starts from scratch, and concretely, but builds abstracion and sophistication throughout the book, until by the end of volume 2 you have left Rudin in the dust.
Could you recommend similar books for me on Algebra and Number Theory? I would also appreciate some comments on the books (including Zorich's one if possible). Some ideas I have after researching:
Number Theory: Hardy/Wright's theory of numbers; Ireland/Rosen Number Theory
Algebra: Cameron's Intro to Algebra (Starts with rings, which I am told is the "right" way to do things); Kostrikin's book of the same name (starts with groups, but has a similar style to Zorich making it very suitable for my purposes)
Analysis: Only Zorich so far. (Note: I was once told that Zorich is like Spivak and Rudin in one.)
To emphasize: The books must go at a fast pace, and take you a long way. For this to work, the books must be concise. "First course" books (such as Burkill's first course in analysis - recommended to Cambridge first year's in the math tripos) don't work for this. They must be books that contain both a "first course" in the subject as well as the later more challenging material towards the end. I hope this has made it clear enough.
Any replies will be very much appreciated
Thank you very much