Beauty in Numbers

Mathematics: The New Golden Age

by Keith Devlin
Penguin, 287 pp., $8.95 (paper)

Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge

by William Poundstone
Doubleday/Anchor, 274 pp., $18.95

The front cover of Keith Devlin’s Penguin paperback is a dazzling computer image reproduced from a Springer Verlag book, The Beauty of Fractals. If it looks familiar it’s because you’ve seen it on the front cover of James Gleick’s recent best seller, Chaos, published by Viking. Two books with identical covers are rare enough, but to make things worse, Viking is now owned by Penguin. I take this blunder to reflect the chaos that develops when big publishing houses merge.

A research mathematician at the University of Lancaster, Devlin is best known in England for his column on mathematics in The Guardian, and for his television appearances. This new book (his ninth) is an effort to introduce general readers to some of the more exciting recent breakthroughs in what he calls “mankind’s most impenetrable subject.” Devlin’s choice of material is excellent, and he is to be praised for the clarity and accuracy with which he presents it.

The book’s first topic is prime numbers: integers greater than 1 that have no divisors except themselves and 1. The largest known prime—it has 65,050 digits—is obtained by raising 2 to the power of 216,091, then subtracting 1. Primes of this form are called Mersenne primes. Only thirty-one are known, and no one has yet proved whether there are infinitely many, or only a finite number, maybe no more at all.

For the past decade there has been an intensive search for faster ways to factor composites—numbers that are not prime. One reason, Devlin tells us, is that improved computer methods of factoring could destroy a widely used cipher known as the RNA system after the initials of its three MIT inventors. It is called a publickey cryptosystem because the method of encoding secret messages can be published and used by anyone. Decoding is something else. It is possible only if one knows the two factors of a gigantic composite, obtained by multiplying two primes, which serves as the cipher’s secret key. When the RNA system was first proposed twelve years ago prime numbers of eighty digits were recommended because there were then no known ways to factor the product of two such primes in “reasonable” computer time—say ten years. The RNA system is still secure, but because of recent improvements in factoring speed, primes of one hundred digits are now used to make two-hundred-digit keys.

Here are quick rundowns on Devlin’s other topics. A chapter on infinite sets includes Kurt Gödel’s famous proof that even in systems as simple as arithmetic there are statements impossible to prove true or false. A discussion of different kinds of numbers centers around an amazing property of 163. A chapter on fractals—structures that always look the same, either exactly or in a statistical sense, as you endlessly enlarge portions of them—introduces the famous Mandelbrot set. It is named after IBM’s Benoit Mandelbrot, who invented the term “fractal …

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