Number Theory

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Now in its second edition, this textbook offers an introduction to number theory focused on the density and properties of prime numbers. This approach provides a solid foundation in standard number theory topics while giving an overview of the discipline. Essential subjects covered include the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New topics in this edition include p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods for primality testing. Key features include a thorough introduction to analytic number theory with complete proofs of Dirichlet's Theorem and the Prime Number Theorem, a concise treatment of algebraic number theory covering primes and unique factorization of ideals, and a discussion of the AKS algorithm demonstrating that primality testing is in polynomial time, a rarity in such texts. Additional topics include primality testing, cryptography, Fermat and Mersenne numbers, and Carmichael numbers. The user-friendly style, historical context, and a wide range of exercises—ranging from simple to challenging (with solutions for select problems)—make this textbook suitable for both self-study and classroom use. It is intended for upper-level undergraduates and beginning graduates, requiring only a basic knowledge of calculus, multivariable calculus, and some linear algebra, with necessary concepts