Many mathematicians believe that one of the things which differentiates mathematics from the other sciences is that our research relies on proofs rather than experiments. While this point of view certainly has some merit, the use of experiments in mathematics has certainly increased in recent decades. As computers have grown stronger and cheaper, the ability of mathematicians to do computations in order to gather data, make conjectures, and (in a limited number of cases) even prove theorems, has grown with them. Nowadays, there are MAA Short Courses, an entire journal, and a large number of books that are dedicated to the experimental side of mathematics.
In his new book, Experimental Number Theory, Fernando Rodriguez Villegas discusses how computers and numerical experimentation can be used in several areas of number theory. The author has not written a number theory textbook — while he often explains terminology and gives theorems that are needed in the work, there are many things that are assumed, and I imagine it would be hard to get much out of this book without the equivalent of a beginning graduate course in number theory. On the other hand, the book is not a manual for PARI/GP — the author's computer package of choice — and while he doesn't assume any familiarity with PARI, a reader who has never used the package before will almost certainly spend some time flipping between the book and the software's documentation. What Villegas has written is a series of case studies in which he shows what computers are (and aren't) able to accomplish in helping number theorists gain insight into the problems they wish to explore.
Some of the topics that Villegas explores using PARI include the number of ways that a given integer can be written as a sum of squares, computing Kronecker characters and local zeta functions, and numerically approximating the densities of primes of a given factorization type as discussed in Tchebotarev's Density Theorem. There is a chapter dedicated to positive definite binary quadratic forms, including GP code to compute class numbers and to look at reciprocity for various quadratic fields. Another chapter looks at partitions of integers and their relationship with irreducible representations of Sn. Yet another deals with p-adic numbers, and explores properties of the p-adic gamma function and Gauss sums. The final chapter is dedicated to polynomials and discusses problems such as finding polynomials with minimal Mahler measure and finding the largest cyclotomic factor of a given polynomial. This is a large number of topics to cover in fewer than 200 pages, and, as mentioned above, Experimental Number Theory is not a book the reader should turn to for their introduction to most of these topics. But for graduate students and faculty who like concrete examples and would like to learn how to play with the more abstract theory in the standard number theory books by authors such as Artin, Serre, and Cassels & Frohlich, Villegas has the answer for you.
As one might expect, Experimental Number Theory is a highly interactive book. In the introduction, the author warns the reader that "reading this book withut a computer at hand would be rather useless," and this reviewer can testify that he is correct. Several times I picked it up on airplanes or in coffeeshops, and I wouldn't get more than a page or two read before I needed to put it down until I had a computer with a copy of PARI handy. Villegas also provides a large number of exercises (many with hints!) so that the interested reader — and it is hard to imagine a reader not being interested after seeing some of the exciting examples contained in the book — can adapt his code to answer different questions or practice the ideas of experimental number theory on their own.
Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include Algebraic Geometry, Number Theory, and Cryptography. He can be reached at firstname.lastname@example.org.