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Background

I am a college pass out. I have done my B.Tech in computer science and masters in quantum information. I am soon going to take up an industrial job. In the free time I got, after passing out from my college till now, ( few months ) I started reading Sipser's book on "Complexity and Computability theory". After reading it I wanted to learn more about it so I turned towards the book by Boaz and Barak on "Complexity theory". When I reached the chapter on "cryptography", I realized I did not know the fundamentals of number theory, so I started reading "Theory of numbers" by Hardy and Wright.

Question

I have read about one-third of Hardy and Wright. It got my really interested in elementary number theory. I am planning to read the complete book and pick it up as my hobby. The problem I am facing is, although till know I am able to follows the proofs on my own, I keep forgetting what I read earlier. For example I am currently reading about continued fractions and approximations of irrationals by rationals. But I don't remember the proofs, tricks, topic etc in detail that I read before it ( although I can always go back and remind myself ). As I read further I only remember the current topic that I read. And I feel if I read the book completely, at the end I would only remember the broad topics and only get a broader view. Also I try to read couple of pages everyday, but there are breaks of long number of days where I go without reading a single page. And when I again start reading I tend to forget even the current topic.

So I wanted to ask what is the correct way to learn number theory ? Should I read Hardy and Wright completely ( I have still to learn of many other topics like pell's equations, diophantine equations etc. ) ? Is there some other more practical way of learning number theory as a hobby ? How do I reach the threshold point of learning elementary number theory ?

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    $\begingroup$ use the book "An invitation to modern number theory" this is very good $\endgroup$ – Dr. Sonnhard Graubner Jan 20 '17 at 9:14
  • $\begingroup$ I believe that learn mathematics is very difficult. I don't know what is your goal learning number theory (and techniques in algebraic and analytic number theory), but it isn't a hobby, from my viewpoint. It is a deep interest, and you will need next years to get abilities. I am agree that you need to do exercises, and see in your library if there is a different book as companion of Hardy-Wright. Maybe a book more thin or with solved exercises. With respect the memory you can make summary records in small pieces of paper, as conceptual records of proofs, examples or techniques. $\endgroup$ – user243301 Jan 21 '17 at 15:40
  • $\begingroup$ Boaz and Barak->Arora and Barak, I believe. $\endgroup$ – O. S. Dawg Jan 23 '17 at 13:43
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As one of my former lecturer's was fond of saying "mathematics is not a spectator sport". By this I mean that if you want to properly learn and understand it then you should get your hands dirty by doing some problems.

This will mean you have to look up definitions often to begin with until they soak in but eventually you also get spot the same tricks being used over and over again as you figure out how to adapt them and will help you see how different topics link together.

You could also try to attempt to prove some of the theorems before reading the proofs in the book as this will have a similar affect but they're not always constructive in teaching you how to actually apply the concept so a few problems to accompany it will still be worthwhile.

Doing it this way will of course take a lot longer but will definitely give you a deeper understanding and the time spent means that hopefully you won't forget all of it when you start the next chapter! Good luck reading.

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I would not recommend continuing with Hardy and Wright. What you need is strength on the basics, not a tourist's view of a wide range of topics. I would focus on problem solving. Do lots of cool problems, taken from various sources, at a level that is reasonably challenging, but not way out of range, so as to develop your problem-solving and proof skills.

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