Selected Number Theory References

General - Elementary Level

[JJ98] Elementary Number Theory by Jones & Jones, Springer-Verlag, 1998

A good general text. Written at a fairly simple level, but covers a broad range of topics in elementary number theory. ($35, UMBC library)

[Silv97] A Friendly Introduction to Number Theory, Silverman, Prentice-Hall, 1997

A good, if somewhat sparse, introduction to number theory. Gives short shrift to some classical topics in order to provide a coverage of elliptic curves. Written with a target audience of non-math majors. ($72)

General - Advanced Level

[Rose96] A Course in Number Theory, 2^nd Ed, by H. E. Rose, Oxford Univ Press, 1996

Good general text, best suited for either advanced undergraduate or first graduate course. ($45, UMCP library)

[NZM91] The Theory of Numbers, 5th Ed, by Niven, Zuckerman & Montgomery, Wiley Publ, 1991

Good text, best suited for either advanced undergraduate or first graduate course. Classical approach with recently added sections having some computational flavor. Previously used as a text for this class. ($95, UMBC library)

[IR91] A Classical Introduction to Modern Number Theory, 2nd Ed, by Ireland & Rosen, Springer-Verlag, 1991

One of the best general number theory books at a graduate level. Little computational coverage but good coverage from a modern algebraic viewpoint. Contains a short introduction to the theory of finite fields. ($60, UMCP library)

[MaWo03] MathWorld, by Eric Weisstein

A comprehensive on-line mathematics encyclopedia, currently hosted by Wolfram at http://mathworld.wolfram.com/topics/NumberTheory.html. Very thorough as a reference, but of no value as a text. Also published in book form by CRC Press.

Computational Methods

[BW00] A Course in Computational Number Theory by David M. Bressoud, Stan Wagon, Springer-Verlag, 2000

A complete text in elementary number theory with a strong computational approach. Uses Mathematica code throughout (sometimes seems to be teaching Mathematica). A potential text for this class. ($65, UMCP library)

[Cohen94] A Course in Computational Algebraic Number Theory by Henri Cohen, Springer-Verlag, 1994

The most thorough and complete reference available on algorithms used in algebraic number theory. A very good reference, but less well suited as a text. ($65, UMCP library)

[BS96] Algorithmic Number Theory - Vol. 1 : Efficient Algorithms by Bach & Shallit, MIT Press, 1996

A coverage of only the polynomial time algorithms (those considered "fast"). A very good reference with nice coverage of complexity questions, but badly needing the promised volume II. Definitely a reference, not a textbook. ($80, UMBC library)

[CP01] Prime Numbers, A Computational Perspective, by R. Crandall & C. Pomerance, Springer, 2001

Another first class exposition of factoring and primality testing/proof methods. Could be used as a text for a second course in number theory, but not a first text. Similar in coverage to [Rie94] but more up to date. ($50, in UMCP libary)

[Kobl94] A Course in Number Theory and Cryptography, by Neal Koblitz, Springer-Verlag, 2^nd Ed 1994

Concentrates on public key cryptography, but provides a solid development of computational number theory in the process. A good textbook for either topic. ($53, 1^st Ed at UMBC library, 2^nd Ed at UMCP)

[Ribe89] The Book of Prime Number Records, by Paulo Ribenboim, Springer, 2^nd Ed 1989

Not a good textbook, but a very good reference, particularly on distribution of primes. 3^rd edition published 1996 under the title New Book of Prime Number Records. ($70, not in UMBC library, 2^nd & 3^rd Ed at UMCP)

[Ries94] Prime Numbers and Computer Methods for Factorization, by Hans Riesel, Birkhäuser, 2^nd Ed 1994

Not a good textbook, but a wonderful reference. Many algorithms are well presented in the form of pseudocode. ($88, 1^st Ed at UMBC library, 2^nd Ed at UMCP)

Number Fields

[Buell89] Binary Quadratic Forms: Classical Theory and Modern Computations, by D. Buell, Springer-Verlag, 1989

Classical (developed by Gauss) theory of quadratic forms, with the modern interpretation as class groups of imaginary quadratic number fields. (UMCP library)

[ST01] Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed, by Stewart & Tall, Peters Publ, 2001

A good coverage of algebraic number fields, finishing with an outline of the proof of Fermat's Last Theorem (admittedly going beyond number fields for that). ($38, 2^nd Ed in UMCP library, 3^rd Ed not in UMd library system)

Finite Fields

[LN97] Finite Fields, Lidl & Niederreiter, Cambridge, 1997

THE reference on the topic. Originally published by Addison-Wesley in 1983, recently republished in a (slightly) revised edition. A cut-down version, recast as a textbook, rather than a reference, was published in 1994. ($130, UMBC library)

[Chil95] A Concrete Introduction to Higher Algebra, 2nd Ed, Lindsay Childs, Springer-Verlag, 1995

A coverage of modern algebra with a strong emphasis on finite, computable structures, in particular those of number theory. A good introductory coverage of finite fields and their applications is in Chap 28. ($53, UMBC library)

Open Questions

[Guy94] Unsolved Problems in Number Theory, 2^nd Ed, R. Guy, Springer-Verlag, 1994

A catalog of open problems, references to work on them and their background. Not something to be read, but used as a reference and guide to the literature. ($50, UMd library, 1^st Ed in UMBC library)

[KW91] Old and New Unsolved Problems in Plane Geometry and Number Theory, V. Klee & S. Wagon, MAA, 1991

Elementary coverage of a short list of classical open problems in number theory. (UMBC library)

[Shpa93] Computational and Algorithmic Problems in Finite Fields, I. Shparlinski, Kluwer, 1993

A catalog of open problems and references. Like [Guy94], this is a reference, not a text. ($200, UMd library)

[Shan85] Solved and Unsolved Problems in Number Theory, 3^nd Ed, D. Shanks, Chelsea, 1985

A coverage of elementary number theory in a non-standard order and approach. Very readable and good pointers to open questions. (UMd library, 1^st Ed in UMBC library)