I'm a high school student, interested in mathematics, especially in number theory. While preparing for the IMO test, and thinking about generalizations or the root of many olympiad problems led me to high level NT.
(for instance, ISL 2011 A8 asks if $x^5 +y^2 $ forms a complete residue system in $mod p$, which is actually killed by Hasse-Weil bound, and a classic problem asking to prove $x^3 -2y^3 =1$ has finite solutions is a special case of Thue's theorem.)
Like this, my NT study has been definitely not systematic. To start with, I've read Analytic number theory(Apostol), and my knowledge in this field is round about understanding the proof of PNT and being familiar with analytically calculating some series related to primes and sieve methods. I've also seen a bit of algebratic number theory. Not much, but I've read the part about quadratic fields in 'Introduction to number theory(Hardy)'. These days, I'm studying about transcendental number theory, and now I'm reading about baker's theorem.
Is my way of studying NT Ok? I feel like I'm studying too randomly, and what I've read doesn't seem to be properly organized inside my brain. Another disadvantage is that studying in this way, I don't see the end;there are so much NT papers, and I feel that I will not be able to cover modern NT even if I study for my whole lifetime.....
Could you give me some advice about how to study NT, and could somebody tell me about roughly how much modern Number theory has developed in this point(i.e. how much should I have to study to study to get on a level to be able to understand most of the papers about pure NT?)