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 Learning 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 [C.1.2], 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.

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.

A quality middle-of-the-road introduction to proof, used reasonably widely and covering all standard topics for a proof transition course.

The title is pretty accurate; this is a quite gentle open text usable for self-study.

This book is not necessarily just an introduction to proof, but has a wonderful attitude to conjecture. Essentially, one should view every proof as an opportunity to extend, and every disproof as a chance to rescue.

This is a very good guide to Sage for anyone starting out with basic college math knowledge; the author has taught using Sage for some time. Did I mention it is freely downloadable as well as available in print?

This guide is not free, but is comprehensive (for the time it was written) and has the unique perspective of someone not involved in the Sage community.

A very good introduction to programming from scratch in Python, usable from the website or as a hard-copy text.

A preternaturally idiosyncratic take on how to program, but well worth the effort to learn things the hard way if you have the time to push through it.

A marvelous achievement of bringing the Riemann Hypothesis to the (determined) lay reader while simultaneously making you care about post-Napoleonic Europe. If I do say so myself.

Interesting lecture notes leading to a basic understanding of the Riemann Hypothesis, based on a high-school enrichment program in the Netherlands.

This book goes straight for the jugular of the Riemann Hypothesis, starting from scratch. That requires a lot of investment, but you won't find it from the perspective of working number theorists in other books, either.

Still useful comprehensive first text on this important topic.

Very innovative book on exactly what it says; second half not necessarily for every US undergraduate, but easiest introduction to Birch-Swinnerton-Dyer I could find! Covers most traditional material, too, and has copious entertaining historical notes.

The canonical “undergraduate” analytic number theory book. Monumental but very difficult; zillions of interesting results in exercises.

Contains Mathematica code to visualize and explore a lot of interesting number theory, and is very consistent with the computational viewpoint throughout.

Definitely a second course in number theory, as the subtitle says, with good material on arithmetic progressions and the Hilbert-Waring problem (the latter is difficult to find in a textbook).

Another well-known general resource, with a very good description of how to find if a rational conic has a rational point (which directly connects to integer points on conics as well).

Many very interesting topics for the general reader, from repunits to all sorts of other topics. Intriguing story must lie behind the essentially identical book by a different author several years later.

The title says it all, and more accessible to college students than one would think. By one of the leaders in the field.

A brilliant, accessible, inventive book which makes me very sad there is only enough time for so many topics in a one-semester course. Indispensable for bringing partitions to undergraduates.

Very conversational and enjoyable; not really a textbook. Key feature is a detailed discussion of how Euler missed what is essentially unique factorization in a certain number field for two of his more interesting results – and he does it without actually proving unique factorization!

This book turns out to be about both \(\Gamma\) the function and \(\gamma\) the constant (recall Definition 20.3.2), and includes a description of Apéry's tomb (see Subsection 24.4.1 and \(\zeta(3)\)).

Delightful introduction to and inspiration for many of the lattice topics pursued in this text. The second half goes fairly deep, and is more than worth pursuing as a directed study with undergraduates.

This book has incredible amounts of interesting detail regarding many of the prime topics considered here. An example: a discourse on whether the pseudoprime criterion base 2 was really discovered by ancient Chinese mathematicians.

Based on a series of lectures, this book is rather higher level, but has correspondingly more truly interesting material, including an entire chapter inspired by \(1093\) and a very early prime-generating algorithm by a certain Pocklington.

Oodles of class-tested historical material and many, many exercises, including a welter of them on topics surrounding amicable numbers.

A very nice historically-oriented approach to elementary number theory. Includes Sage material in an appendix.

Another conversational classic by Ore, with plenty of historical goodies.

A truly historical sojourn through much of number theory up through the early twentieth century, with extensive primary source material and investigation of Gauss' monumental work. Sadly, largely beyond the level of this text.

This is intended for those without calculus, but has many great number-theoretic bits all the same.

This book has some nice discussion of Euler's number theory alongside many other historical vignettes with real math power.

Collection of additional classroom resources focused on primary source material, including the Basel problem and quadratic reciprocity.

This delightful picture book has a different monster for each prime number, with bizarre combinations for composites. Personal experience says it satisfies for ages three and up.

Visualize group theory; gorgeous pictures.

A joyous and pictorially engaging romp.

A very good compendium of many articles (published throughout the years) most appropriate for teachers of undergraduate number theory.

Aimed at bringing number theory to in-practice or pre-practice educators, this has a very nice treatment of arithmetic functions. Once you've heard of summation and Moebius inversion as ‘parent’ and ‘child’ relationships, you'll never think of them the same again.

Aimed at bringing algebra and geometry to in-practice or pre-practice educators; manages to bring Gaussian and Eisenstein integers and some quadratic forms in at the ground level.

The subtitle is “and other discrete mathematical adventures”, and that about says it. Covers a surprising amount of number theory in very visual ways.

Unfortunately no longer in print, but a very good source of ideas for connecting what we usually think of as the continuous world of calculus and various discrete topics (not just number theory, though this shows up in several chapters).