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I'm currently trained in Algebra, Calculus and Statistics in high school level. Basically, I've no knowledge at all in number theory as this subject is not taken seriously in my country.

I was thinking to self-learning number theory by books. But the books I found in the internet are really hard and complicated for me as I've no number theory background at all.

Recently, I was asked by a 10 years old kid for a question which is related to number theory. I've no idea at all how to solve that, it was really embarrassed.

So can anyone recommend me some books for number theory which are suitable for me?

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  • $\begingroup$ A short, easy book on number theory at the high school level is Invitation to Number Theory by Oystein Ore. $\endgroup$ – user49640 Jun 7 '17 at 8:51
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The greatest of all classical books on this subject is An Introduction to the Theory of Numbers, by G. H. Hardy and Edward M. Wright.

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The Higher Arithmetic: An Introduction to the Theory of Numbers by Harold Davenport

Introduces the classic concepts and theorems of number theory in a way that does not require the reader to have an in-depth knowledge of the theory of numbers

and Elementary Number Theory by Sierpinski

The variety of topics covered (...) includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers.

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  • $\begingroup$ I think Davenport's book covers great introductory topics and contains many gems, but the style of the book is not well suited to a first-time student (surprising, given the title and aim). In some proofs the choice of which steps are detailed and which are omitted leads to more obfuscation than clarity for a first-time student (see the proof of quadratic reciprocity). The new student hopes to gain a better "feel" for the subject with such a conversational style, but I think that for every point where this succeeds, it causes enough trouble in other places that ultimately it is not worth it. $\endgroup$ – Jonathan Rayner Jul 8 at 14:46
  • $\begingroup$ To provide a counterpoint to my own comment: Davenport is incredible for an early student to learn "about" Number Theory as a field - it does an excellent job of mentioning "big" open problems that have simple formulations, the history and motivating questions that lead to be more abstract modern formulations, and so on. These are the types are things that are exciting for first-time students and that is a good strength. $\endgroup$ – Jonathan Rayner Jul 8 at 14:51
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A Classical Introduction to Modern Number Theory by Ireland and Rosen.

Fantastic undergraduate book that covers a lot of ground. While it doesn't require much background, many of the proofs are terse and the authors expect a lot out of you. But it's a very rewarding read and in addition to the number theory you will learn, this book will greatly improve your ability to read mathematics.

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For Elementary number theory, I would recommend David M. Burton's Elementary number theory and for a bit higher studies my choice is An introduction to theory of numbers by Niven, Zuckerman, Montgomery

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  • $\begingroup$ Is this link to an illegal copy? $\endgroup$ – lhf Jun 7 '17 at 9:30
  • $\begingroup$ @lhf Well in my opinion no!! Is there any problem?? I m using this pdf from long time $\endgroup$ – Vidyanshu Mishra Jun 7 '17 at 9:40
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I would recommend David my Burton elementary number theory. Most mathematicians prefer it for starting.A very concise and at the same time complete book

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  • $\begingroup$ Is this link to an illegal copy? $\endgroup$ – lhf Jun 7 '17 at 9:30
  • $\begingroup$ I don't think so. It is an easily available pdf version. $\endgroup$ – vutsuak Jun 7 '17 at 9:33
  • $\begingroup$ Simon Plouffe is a very prominent mathematician (of 'Borwein Bailey Plouffe' fame, if that rings any bells) and it seems highly unlikely (though admittedly not impossible) that he'd be hosting an illegal copy. $\endgroup$ – Steven Stadnicki Jun 7 '17 at 19:02
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I had good experience with the Number theory book by Borevich and Shafarevich.

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Another good option is:

B. Fine and G. Rosenberger, Number Theory: An Introduction via the Density of Primes

From Amazon review: "All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes."

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Im my option the best way to start learning this subject would be "Elementary Number Theory" by Jones & Jones

As far as I can remember this has prequisites really.

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