Number Theory: In Context and Interactive

A good introduction with an emphasis on groups, containing interleaved exercises with full answers.

A highly regarded text with copious notes, but sometimes more than a little hard to parse with its consecutively numbered theorems and very dense prose.

Freely available and the first Sage-enabled number theory text, by the founder of Sage (a number theorist).

A venerable text with programming exercises that still wear well.

The topics are very standard, but the approach is quite different; no proofs, only statements. This turns out to be a highly effective pedagogy; see the Academy of Inquiry Based Learning⁶ for more information.

A very fun inquiry-driven text before there were such things, with a lot of extremely good examples, especially in things like quadratic forms.

More algebraically oriented, with good material on the Pell equation and Gaussian integers – noteworthy for a good treatment of Conway’s river concepts.

Incredibly comprehensive, at a fairly high level. Good material on averages and odd perfection, immense bibliography and notes in style of [E.2.2], and also inquiry-driven “do-it-yourself” sections. Appears to be out of print.

Surprisingly detailed and high-level but has good coverage of several unusual topics such as geometry of numbers.

Enough material for two courses, some fairly advanced, and newly endowed with downloadable Sage worksheets for use with local or online CoCalc¹⁰.

Yet another nice reprint from Dover, this one with (as one would expect of the author) great combinatorial content.

Not so algorithmic, but very, very concrete and constructive. Squares are \(\square\)s, which grows on the reader.

An out-of-print standard text with many similar topics and interesting historical comments.

An older text that is still worth the conversational tone.

Careful emphasis throughout on getting a novice student ready for abstract algebra/algebraic number theory, with \(\mathbb{Q}[\sqrt{d}]\) coherent in an elementary text. Don’t miss continued fractions in the service of the Bezout identity and the many interesting projects, including one on the \(p\)-adic numbers.

See my review for MAA reviews¹⁶ of this relatively ambitious text. Could be very interesting to use for a two-semester algebra sequence that starts with number theory.

An inquiry-friendly introduction with a uniquely elementary tilt toward algebraic number theory. And many, many puns.