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Number Theory: In Context and Interactive

Karl-Dieter Crisman

References E.4 Specialized References

Number Theory is a huge field, and even at an introductory level there are many wonderful resources to be aware of. I have used many of the following in one way or another in preparation of this text, and if you are intrigued by a specific facet of number theory, I encourage you to get these from your library! Most of these are more specialized, but a few are not really texts but intended for the “casual” reader.
[1]
John Derbyshire, Prime Obsession, Joseph Henry Press, (2003) (Website
 27 
www.nap.edu/catalog/10532/prime-obsession-bernhard-riemann-and-the-greatest-unsolved-problem-in
)
Note.
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
 28 
www.booksandculture.com/articles/2009/janfeb/prime.html
.
[2]
Roland van der Veen and Jan van de Craats, The Riemann Hypothesis, Mathematical Association of America, (2016). (Website
 29 
www.maa.org/press/books/the-riemann-hypothesis
)
Note.
Interesting lecture notes leading to a basic understanding of the Riemann Hypothesis, based on a high-school enrichment program in the Netherlands.
[3]
Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, (2016). (Website
 30 
wstein.org/rh/
)
Note.
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.
[4]
H. M. Edwards, Riemann’s Zeta Function, Dover, (2001) (Website
 31 
store.doverpublications.com/0486417409.html
)
Note.
Still useful comprehensive first text on this important topic.
[5]
Jeffrey Stopple, A Primer of Analytic Number Theory, Cambridge, (2003). (Website
 32 
www.cambridge.org/us/academic/subjects/mathematics/number-theory/primer-analytic-number-theory-pythagoras-riemann
)
Note.
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.
[6]
Tom Apostol, Introduction to Analytic Number Theory, Springer, (1976). (Website
 33 
www.springer.com/us/book/9780387901633
)
Note.
The canonical “undergraduate” analytic number theory book. Monumental but very difficult; zillions of interesting results in exercises.
[7]
Stan Wagon and David Bressoud, A Course in Computational Number Theory, Wiley, (2008). (Website
 34 
www.wiley.com/WileyCDA/WileyTitle/productCd-0470412151.html
)
Note.
Contains Mathematica code to visualize and explore a lot of interesting number theory, and is very consistent with the computational viewpoint throughout.
[8]
Paul Pollack, Not Always Buried Deep, American Mathematical Society, (2009). (Website
 35 
bookstore.ams.org/mbk-68
)
Note.
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).
[9]
Şaban Alaca and Kenneth S. Williams, Introductory algebraic number theory, Cambridge University Press, (2003). (Website
 36 
www.cambridge.org/core/books/introductory-algebraic-number-theory/9F53B233CD4D717B1A31ECD117FFEA7D
)
Note.
As the title says, and one appropriate for an undergraduate library.
[10]
Harold Davenport, The Higher Arithmetic, Cambridge University Press, (2008). (Website
 37 
www.cambridge.org/us/academic/subjects/mathematics/number-theory/higher-arithmetic-introduction-theory-numbers-8th-edition
)
Note.
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).
[11]
Stephen Richards, A Number for Your Thoughts, S. P. Richards, (1982) (No website)
Note.
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.
[12]
Samuel S. Wagstaff, Jr., The Joy of Factoring, American Mathematical Society, (2013). (Website
 38 
bookstore.ams.org/stml-68
)
Note.
The title says it all, and more accessible to college students than one would think. By one of the leaders in the field.
[13]
George Andrews and Kimmo Eriksson, Integer Partitions, Cambridge University Press, (2004). (Website
 39 
www.cambridge.org/us/academic/subjects/mathematics/number-theory/integer-partitions
)
Note.
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.
[14]
Richard Friedberg, An Adventurer’s Guide to Number Theory, Dover, (1995) (Website
 40 
store.doverpublications.com/0486281337.html
)
Note.
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!
[15]
Julian Havil, Gamma: Exploring Euler’s Constant, Princeton, (2009). (Website
 41 
press.princeton.edu/titles/7494.html
)
Note.
This book turns out to be about both \(\Gamma\) the function and \(\gamma\) the constant (recall Definition 20.3.10), and includes a description of Apéry’s tomb (see Remark 24.4.1 with regard to \(\zeta(3)\)).
[16]
C. D. Olds, Anneli Lax, Giuliana Davidoff, The Geometry of Numbers, Mathematical Association of America, (2000) (Website
 42 
www.maa.org/press/ebooks/the-geometry-of-numbers
)
Note.
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.
[17]
Paulo Ribenboim, The Little Book of Bigger Primes, Springer, (2004) (Website
 43 
www.springer.com/us/book/9780387201696
)
Note.
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.
[18]
Paulo Ribenboim, My Numbers, My Friends, Springer, (2000) (Website
 44 
www.springer.com/us/book/9780387989112
)
Note.
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.
[19]
Thomas R. Shemanske, Modern Cryptography and Elliptic Curves: A Beginner’s Guide, American Mathematical Society, (2017) (Website
 45 
bookstore.ams.org/stml-83
)
Note.
This really is a beginner’s guide, which developmentally arrives at addition on projective elliptic curves. The focus on cryptography is clear with Lenstra’s ECM algorithm as payoff, but BSD is also reasonably described. But why mention safe primes and not Germain primes?
[20]
Martin H. Weissman, An Illustrated Theory of Numbers, American Mathematical Society, (2017), (Website
 46 
illustratedtheoryofnumbers.com
)
Note.
Lushly illustrated, including for nonstandard topics like Conway’s topograph and Gaussian/Eisenstein. Emphasis on dynamical point of view, even for Euler’s Theorem. Well-researched historical notes, and linked Jupyter notebooks on the website.
[21]
Benjamin Hutz, An Experimental Introduction to Number Theory, American Mathematical Society, (2018), (Website
 47 
bookstore.ams.org/amstext-31
)
Note.
Many in-depth topics somewhat beyond a standard semester course, such as height and Diophantine approximation. Unique is covering dynamical systems on polynomials over \(\mathbb{Q}\text{.}\) The intriguing exploratory exercises lack pseudocode.
[22]
Alasdair McAndrew, Introduction to Cryptography with Open-Source Software, CRC, (2011), (Website
 48 
www.crcpress.com/Introduction-to-Cryptography-with-Open-Source-Software/McAndrew/p/book/9781439825709
)
Note.
I have not read this, but with full sections on DES and AES, elliptic curves, and “El Gamal in Sage”, I think it could be a good complement on the application side to many of the texts in these references.
[23]
Avner Ash and Robert Gross, Fearless Symmetry, Princeton, (2008), (Website
 49 
press.princeton.edu/books/paperback/9780691138718/fearless-symmetry
)
Note.
Astonishingly, builds up in a conversational tone from practically nothing to Galois representations coming from elliptic curves and the connection to Fermat’s Last Theorem. Explicitly connects quadratic reciprocity to quadratic curves, for instance. Highly recommended.
[24]
Avner Ash and Robert Gross, Elliptic Tales, Princeton, (2012), (Website
 50 
press.princeton.edu/books/hardcover/9780691151199/elliptic-tales
)
Note.
A followup to [E.4.23], which attempts to explain elliptic curves from the ground up through to their \(L\)-functions and the Birch-Swinnerton-Dyer conjecture.
[25]
Lasse Rempe-Gillen and Rebecca Waldecker, Primality Testing for Beginners, American Mathematical Society, (2014), (Website
 51 
bookstore.ams.org/stml-70
)
Note.
Although it does cover a lot of basic number theory, the unusual main focus is making the proof of Agrawal, Kayal, and Saxena that deciding whether a number is prime is in the computational complexity class \(P\) directly accessible to (talented) high school and university students.
[26]
Paul Pollack, A Conversational Introduction to Algebraic Number Theory, American Mathematical Society, (2017), (Website
 52 
bookstore.ams.org/stml-84
)
Note.
Definitely requires a good ring and field background, but also truly conversational. It starts with a very thorough treatment of quadratic number fields, then starts over, meanwhile making reference to a startling number of both original papers from the nineteenth century and very recent Monthly articles.
[27]
Roger Plymen, The Great Prime Number Race, American Mathematical Society, (2020), (Website
 53 
bookstore.ams.org/stml-92/
)
Note.
A great deal of information about the zeta function, especially the functional equation, with an eye toward both the explicit formulas and specifically Littlewood and Skewes’ results.
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