Gelfand: The Solution of Equations in Integers -A good review of some of the topics covered this summer plus a glimpse of what lies beyond. Includes a treatment of cubic and biquadratic reciprocity.
Ireland and Rosen: A Classical Introduction to Modern Number Theory - Covers the rudiments of algebraic number theory concretely using mostly elementary methods. Hardy and Wright: An Introduction to the Theory of Numbers - A thorough and classical treatment. Contains a surprising wealth of information.
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Also serves as an excellent tutorial on how to read harder math books in general. Now reprinted by Dover, so you have no excuse for not owning it. M.: Riemann’s Zeta Function - A great description of the zeta function and what is known about it assumes complex analysis. Davenport: Multiplicative Number Theory – Good introduction to Dirichlet’s theorem, the zeta function, and sieve theory. Davenport: T he Higher Arithmetic, 7th Ed., Cambridge - A good book to read after PROMYS. Cox, David: Primes of the form x^2+ny^2 - A wonderful book that introduces such ideas as complex multiplication, class field theory, elliptic curves, and binary quadratic forms, in a down to earth and very accessible way! Lots of fun. Includes unusual proofs of Hasse-Minkowski, quadratic reciprocity, and the three squares theorem. Conway, John H.: The Sensual (Quadratic) Form - A unique approach to quadratic forms. It is a good book to read after PROMYS, especially if Z upsets you. Cohn: Advanced Number Theory - This book, as its name suggests, is fairly advanced but it is quite accessible. Cohen: A Course in Computational Algebraic Number Theory - Your number one source for algorithms in computational algebraic number theory. Hard to read without a strong background in such things. Borevich & Shafarevich: Number Theory - A beautifully written “introduction” to the more advanced aspects of modern algebraic number theory. Some familiarity with calculus is assumed and the last few chapters assume some knowledge of complex analysis. Apostol Introduction to Analytic Number Theory - A good introduction to both analytic number theory and to many topics which the PROMYS number theory covers lightly. Andrews: Number Theory - A good introductory text with some unusual combinatorial proofs.
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