(I'm not myself capable of doing research in number theory yet, but I have some knowledge of what it takes as I'm considering specializing in it. Take my advice with a grain of salt as I may not really have the authority to speak on this matter.)
It depends on the kind of research. Number theory is a huge field. After reading and fully understanding the three number theory books you may be able to understand a couple of research papers on number theory (mostly old ones), but to be blunt 3 books will not be anywhere near enough to do serious research in modern number theory. In fact most of the contents of these books can be covered in an undergraduate elementary number theory course. Number theory is a kind of peculiar field and even today sometimes elementary proofs of elementary results are published, but most of the researchers doing this have a firm foundation in mathematics and number theory from where they draw inspiration.
Mathematics students use 7+ years of full-time study guided by professors before they are able to take on original research, and even then it's usually guided by a professor. Just like there is no royal road to geometry, there is no royal road to number theory. You have to learn it the hard way and that takes years of dedication. Mathematics is highly interconnected and you cannot just ignore large parts of it.
Number theory is a diverse field and you do not need to understand it all, but if for instance you want to understand algebraic number theory you should have a decent grounding in algebra (at least at the level of a serious graduate book like Lang's Algebra), and a couple of graduate books on algebraic number theory (algebraic number theory by Lang is a good introduction, and most books by Jean-Pierre Serre are good follow-ups). This should prepare you to study the prerequisites for a more specific area for which you will need a couple of more graduate level books (maybe you want to focus on certain kinds of cohomology in number fields, or global class field theory). Similar roads are needed for other kinds of number theory. For instance in analytic number theory you will need a solid grounding in analysis as well as a couple of graduate books on analytic number theory.
Of course while reading these books you will find that to understand them, and more importantly to understand their motivation, you need to read up on other mathematical topics such as Galois cohomology, diophantine geometry, algebraic topology, homological algebra, etc.
Once you have done all this you may be able to understand the technical details in number theory papers, but of course you will not have the experience yet to perform original research. For this you will likely need more perspective from talking to active researchers, reading articles, and studying related parts of mathematics.
To give you an extent for the prerequisites let me quote from "Algebraic Number Theory" by Serge Lang which is supposed to be an introduction to algebraic number theory:
Chapters I through VII are self-contained, assuming only elementary algebra, say at the level of Galois theory.
Some of the chapters on analytic number theory assume some analysis. Chapter XIV assumes Fourier analysis on locally compact groups. ...
Galois theory, and Fourier analysis on locally compact groups are pretty heavy prerequisites for most CS students. Especially in an introductory text.
You will basically need the equivalent of a phd in math to actively do research in number theory. I'm not trying to discourage you as math and number theory are beautiful subjects worthy of studying even if you do not reach the research level, but assuming those books are enough is like assuming reading "An introduction to algorithms" (CLRS) is enough to do independent modern CS research in the field of algorithms. It's only an introduction that prepares you for much deeper treatments (but CLRS is enough to understand many applications of algorithms outside of CS).
You should know however that modern number theory isn't what is used in most computer science applications. Most of the computer science applications are of elementary number theory of the type covered in Niven. Even when more advanced number theory is used it's usually in the form of a theorem that looks elementary, but where the only known proofs uses advanced number theoretic machinery.
Consider that pretty much every graduate student in mathematics has the knowledge equivalent to what is obtained from your books, and they are all looking for potential research topics. In addition they have professors to help guide them, and who themselves are looking for worthwhile research to do. And these people have a lot more related knowledge and perspective (which helps a lot in math). You may spot what they do not and come up with an elementary proof for a remarkable result, but the odds are not good until you get to much more specialized fields and gain knowledge comparable to the other researchers.
Most of the low-hanging fruit has already been grabbed in number theory as it's such an old subject, and the only fruit (problems) left needs very elaborate arguments and heavy machinery. Of course once in a while someone stumble upon a single fairly elementary problem, but it becomes rarer as the subject matures and more people have looked for these problems.
