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Elementary number theory. Primes, congruences, and secrets.


“Oh no, not yet another introduction to elementary number theory!” - Franz Lemmermeyer's review of William Stein's “Elementary number theory. Primes, congruences, and secrets. A computational approach” [Undergraduate Texts in Mathematics. New York, NY: Springer (2009; Zbl 1155.11002)] starts with irony, but comes quickly to its advantages:

“One cannot expect fantastic new proofs in an area which has been covered in countless textbooks, and so such a book has to be judged by its exposition. In this respect, the author succeeds admirably: the book is written in an entertaining style; it even contains a few jokes, such as Lenstra's proof that there are infinitely many composite numbers, and subjects that are usually quite dry are presented in a lively way.” [..]

“What distinguishes this text from other books is the computational approach, which the author takes seriously. He uses the software system Sage developed by himself throughout the book, from computing greatest common divisors to proving the associativity of the group law on an elliptic curve, or for determining its rank. It doesn't take much to predict that software systems will play an ever increasing role in the future of mathematics, and having a text explaining a powerful system such as Sage has two advantages: it gives the students a tool to do calculations that illustrate even the most abstract concepts, and, simultaneously, introduces them to an open source software that can later be applied profitably for studying research problems.” [..]

“Yet another introduction to number theory? Yes, but an excellent one, with the additional bonus of a) presenting material that can be covered in one semester, and b) introducing the readers to a powerful software system.”

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Abel prize 2010
I. M. Gelfand 1913-2009

Scientific prize winners of the ICM 2010
Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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