João da Silva is trained as an electrical engineer and currently works as a cryptography consultant. His book on the prime numbers is an enthusiastic popularization directed to a general reader who has at least a background in high school algebra. The author shows very decent skills in exposition, particularly as he works his way through arguments and proofs that might otherwise be intimidating to an unsophisticated reader. The tone throughout is conversational — as if he were across the room telling us “look at these wonderful things I’ve found.”
Although prime numbers are the focus, the author’s breadth of interest is large and ranges from theoretical physics (the p-adic numbers and cosmology) to factoring, cryptography, the Goldbach and twin primes conjectures and the Collatz problem. Problems are interspersed throughout the text and complete solutions are provided in an appendix. The author prefers an experimental approach, and he relies heavily on Mathematica as a convenient tool. All the Mathematica code for functions used throughout the book is provided in an appendix (for the reader who might be interested), but its explicit role in the text is quite minimal.
I was charmed by the approach the author describes to proving the Goldbach conjecture. He sets out his plan, carries it out and then discusses why it fails and what one might do to improve the approach. Is there a better way to describe mathematics in action, as a living creation, to those who might believe it’s all known and recorded in textbooks?
The Collatz problem (aka the 3x + 1 problem, or Kakutani’s problem, or Ulam’s problem, or ...) is taken up in another chapter. It does not have an immediate or obvious connection with prime numbers, but da Silva treats it in a novel and interesting way.
Only a modest part of the book is given to the author’s area of specialization in cryptography. He discusses the RSA algorithm and the notion of codes based on the difficulty of factoring composite numbers with large prime factors. It would have been interesting to get his sense of the vulnerability of these algorithms to increasingly powerful factoring methods.
The book’s title derives from a chapter on the visual representation of primes, and — in particular — the fractal binary pattern that he associates with the primes. There are several colorful diagrams, and it is clear that this is one of the author’s major interests, but it is the book’s weakest chapter.
There are occasional, relatively rare, infelicities. For example, after writing out the equation of Fermat’s last theorem, da Silva writes “...where n is not valid for n > 2...” The choice of topics is somewhat eccentric, but clearly driven by the author’s enthusiasms.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.