Which is the single best book for Number Theory that everyone who loves Mathematics should read?
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I recommend Primes of the Form x2 + ny2, by David Cox. The question of which primes can be written as the sum of two squares was settled by Euler. The more general question turns out to be much harder, and leads you to more advanced techniques in number theory like class field theory and elliptic curves with complex multiplication.
One of my colleagues, a number theorist, recommended the little book by van den eynden for beginners. my favorite is by trygve nagell. (I am a geometer.) One of my friends, preparing for a PhD in arithmetic geometry?, started with the one recommended by Barry, Basic number theory. As I recall it's for people who can handle Haar measure popping up on the first page of a "basic" book on number theory.
I also recommend Gauss's Disquisitiones Arithmeticae.
A Friendly Introduction to Number Theory by Joseph H. Silverman. Although the proofs provided are fairly rigorous, the prose is very conversational, which makes for an easy read. Also, the material is presented so that even a student with a low to moderate level of mathematical maturity can follow the text conceptually and do many of the exercises, but there are plenty of exercises to stretch the more curious mathematician's mind.
As an undergrad I found it very useful and even years later it is one of my all-time favorite number theory references.
It depends on the level.
For an undergraduate interested in algebraic number theory, I would strongly suggest (parts of) Serre's Cours d'arithmetique and also Samuel's Théorie algébriques des nombres.
For a graduate student aiming at a future of research work in number theory, Cassels & Fröhlich is a must.
Another interesting book: A Pathway Into Number Theory - Burn
[B.B] The book is composed entirely of exercises leading the reader through all the elementary theorems of number theory. Can be tedious (you get to verify, say, Fermat's little theorem for maybe $5$ different sets of numbers) but a good way to really work through the beginnings of the subject on one's own.
Number Theory For Beginners by Andre Weil is the slickest,most concise yet best written introduction to number theory I've ever seen-it's withstood the test of time very well. For math students that have never learned number theory and want to learn it quickly and actively, this is still your best choice.
For more advanced readers with a good undergraduate background in classical analysis, Melvyn Nathason's Elementary Number Theory is outstanding and very underrated. It's very well written and probably the most comprehensive introductory textbook on the subject I know,ranging from the basics of the integers through analytic number theory and concluding with a short introduction to additive number theory, a terrific and very active current area of research the author has been very involved in.I heartily recommend it.
For people interested in Computational aspects of Number Theory, A Computational Introduction to Number Theory and Algebra - Victor Shoup , is a good book. It is available online.
William Stein has shared his Elementary Number Theory online: http://wstein.org/ent/ It is accessible, lots of examples and has some nice computation integration using SAGE. I'll be using it this semester with secondary teachers, and will report back if things go particularly well or poorly with it.
Perhaps "best ever" is putting it a bit strong, but for me one of the best besides L E Dickson's books was "Elementary Number Theory" by B A Venkov, which does have an English translation.
One advantage of this book is that it covers an unusual and quite eclectic mix of topics, such as a chapter devoted to Liouville's methods on partitions, and some of these are hard to find in other texts.
The best benefit for me, paradoxically, was that the English translation I worked with was littered with misprints, in places a dozen or more per page. So after a while it became quite an enjoyable challenge to find them, and this meant having to study and consider the text more closely than one might have done otherwise!
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