I am not an official number theorist, like most in the target readership of this intriguing book, but I do belong to the set of “math enthusiasts of all backgrounds” for whom this book was written.

The book’s introduction guides us into its subject, namely the consequences and ramifications of the purportedly “primitive or easy” operation of integer addition--and gives us the option of either “skip[ping] this section [introduction]” or “mus[ing] a little on philosophical problems connected with the concepts and practice of mathematics, as exemplified in ... this book.” I strongly advocate the latter, that is, not skipping, as the introduction summarizes with great clarity the issues at the foundation of mathematics with which the ancient Greeks, Kant, Russell, Wittgenstein, and Quine (among others) grappled. For example, Kant regarded 2+3=5 as a synthetic a priori truth that is independent of experience, whereas this book puts Quine as questioning Kant’s (and subsequent others’) analytic-synthetic dichotomy “entirely.”

The first part (of three) of the book does not require any calculus, and deals with such things--in the realm of the discrete--as formulas for the sum of nth powers of integers. The motivation is that such formulas “lead to beautiful methods of inquiry and amazing proofs.” The second part requires some calculus as it deals with infinite series and how the latter inform the powerful notion of generating functions. The final part, which deals climactically with modular forms, closes (in my words) the number-theoretic loop with respect to the book’s subject, namely integer addition’s far-reaching harvest of techniques and results.

I state the obvious in asserting that the 20th century saw at least two world-shaking mathematical results: Gödel’s 1931 proof of arithmetic’s (and other formal systems’) incompleteness, and Wiles’ 1993-1995 proof of Fermat’s last theorem (FLT). The third part of the book, for which the authors skillfully prepare the nonspecialist reader, treats modular forms, which are key ingredients in Wiles’ 100-page proof of FLT. Though the 1931 reports of the demise of mathematics were, in Mark Twain’s words, highly exaggerated, I admit to retaining sadness (and infantile anger!) regarding the existence of number-theoretic propositions that cannot be reached syntactically from, for example, Peano’s very clean axioms. It is safe to say that Wiles’ proof is not obviously a progression of steps that begin with Peano; the proof seems so far, and to my nonspecialist mind, to be “the only game in town” that is based on conventional, albeit ingenious in the extreme, mathematics. This book takes you for a most exhilarating ride to topics and techniques that perhaps you thought beyond your nonspecialist reach. (A seemingly orthogonal route, quite ingenious in itself, is Professor Harvey Friedman’s “reverse mathematics,” which searches for axioms that would entail a given statement, such as that of FLT.)

The subject matter of the first, non-calculus part of the book is certainly about the integers, but brings the word “enrichment” to mind: congruences and Wilson’s “striking” theorem regarding (p-1)!, with p a prime; quadratic residues and Legendre notation; and sums of two, three, and four squares, with p = 1 mod 4 implying that p is the sum of two squares. These are jewels to which many of us have been exposed. And then there is the “normative” (my word) lesson: “There are ... statements in number theory that are true for billions of examples but turn out to be provably false.” An opportunist like me will turn this into Dijkstra’s “testing reveals the presence of bugs, but never their absence”--which is a lesson yet to be learned by managers of software-intensive projects. Part 1 continues on to sums of higher powers, using both little and “lots of” algebra.

Part 2 prepares us for Part 3’s modular forms and is about infinite series; complex numbers and the complex (upper-half) plane; analytic continuation; the gamma and Riemann zeta functions; non-Euclidean, hyperbolic geometry; the variable q = exp(2&pgr;i z), as it will be used in modular forms; and Bernoulli numbers, as they are surprisingly connected with the zeta function.

Part 3, “Modular Forms and Their Applications,” is in my (stolen) words all you have ever wanted to know about modular forms and FLT but were afraid to ask (as Wiles’ proof is 100 pages long). “Form” in number theory characterizes a function under change of variable. Group theory figures large in these considerations. I’m also glad, on a very mundane level, that the authors characterize a function as a “rule,” and that professional mathematicians can (again) use terms like “source” and “target” instead of “domain” and “co-domain.” (Regarding the latter, it took time for me to be disabused of using “range.”)

I did not expect as a nonspecialist to derive as much from this excellent book as I did, and intend to fill in more gaps upon rereading it--which I deem very much worthwhile.

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