I am a first year undergrad and have had elementary course in Number Theory which includes only basic introductory topics like: divisibility,gcd-lcm,primes,congruences, number theoretic functions etc.
Based on my experience in this subject, I have decided to study it further on my own.
I am planning to cover these topics: Primitive roots,Quadratic residues,binary quadratic forms, etc...

Can someone tell me in what sequence should I study these topics?
Though I have seen that in most books Primitive roots comes just before Quadratic Residues follwed by Binary quadratic forms ,And what other topics follow these three topics?

Second, What I want to have is knowledge in my favourite field of maths which is number theory, Is my approach to learn this is right? Or, should I first see which branch : Algebraic or Analytic interests me more and study only that particular branch?

And most important, Which books or other references like lecture videos should I refer to, suitable to learn the material step by step with exercises to work out?
What opinion you guys have about David Burton book and Alan Baker book?

I apologize if a similar question has already been asked by someone or if I haven't added appropriate tags or If I have asked too many questions in just one post.

And please do not recommend books like: introduction to the theory of numbers by Hardy and wright and another one by Niven, Zuckerman, their are- not at all suitable for someone studying the topics for the first time.
Thanks in advance to all.

  • $\begingroup$ Tom Apostol's Introduction to Analytic Number Theory is a good read and is a 'smooth ramp' to get people started. $\endgroup$ – mathematics2x2life Jan 24 '14 at 7:23
  • $\begingroup$ @BalarkaSen It mentioned in the middle Algebraic/Analytic. With knowledge common to an undergraduate, his book is a nice place to start and covers a nice amount of Analytic Number Theory along with some things of interest to both Algebraic and Analytic Number Theorists. Most people wouldn't have the algebra background necessary to start with introductory books on ANT like Borevich and Shafarevich, Serre, or Langs book. I don't know of any undergraduate ANT theory book. So I recommended the Analytic side. Others can fill in my gap for the introductory algebraic side. $\endgroup$ – mathematics2x2life Jan 24 '14 at 7:29
  • $\begingroup$ @BalarkaSen Yes, I agree that is a good book as well. I just don't tend to think of it as an ANT book so it didn't come to mind. I don't know why. If the OP is looking for books along those same lines, I learned some of the topics covered in that book in Rotman's Advanced Modern Algebra, though it's a long read and obviously its focus is not solely ANT. $\endgroup$ – mathematics2x2life Jan 24 '14 at 7:37
  • $\begingroup$ I would also argue that Hardy's book is suitable for a beginner. $\endgroup$ – finitud Oct 30 '14 at 9:28
  • $\begingroup$ @finitud: Hardy's book is a good book but not particularly suitable for a beginner. $\endgroup$ – user 170039 Jan 24 '15 at 6:25

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