The Music of the Primes

Searching to Solve the Greatest Mystery in Mathematics

by Marcus du Sautoy

348 pages, ISBN 0-06-093558-8, HarperCollins, New York, 2003.

www.harpercollins.com

Reviewed by J. M. Haile, Macatea Productions, http://www.macatea.com/

Ok, can you connect all these: the sieve of Eratosthenes, Goldbach's conjecture, imaginary numbers, Turing Machines, public-key encryption, Fry's Electronics, and 42? (Yes, that 42: the answer to Life, the Universe, and Everything.) Marcus du Sautoy connects these together with much, much more in this extraordinary history of attempts to prove the Riemann Hypothesis about prime numbers.

The primes are tantalizing, enigmatic, inscrutable. Count to 100 by tens, or fives, or even sevens, and you find a pattern; count out loud and you can hear a pattern. Repeat the counting a few times, and you can infuse the pattern with rhythm: SEVEN, 14, 21, 28, THIRTY-five, 42, 49, 56, SIXTY-three, 70, … But try counting to 100 by primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 … It's hard to get your mind around the sequence, for there is no pattern. With practice you can learn the 25 primes below 100, but it's still hard to attach a rhythm to a sequence that has no pattern. Riemann's hypothesis postulates that the sequence of primes does have a pattern, albeit an abstract one. The mystery is whether this true, for it remains unproven.

Is this important? Aren't the primes just a mathematical curiosity? Hardly. First, the primes are the building blocks for all other numbers: the primes are to mathematics as the periodic table of the elements is to chemistry. And du Sautoy claims that chemists know much more about the elements than mathematicians know about prime numbers.

Second, in the 147 years since Riemann first published his hypothesis, a substantial body of mathematics has been constructed on the presumption that it is true. If it is not, then many "theorems" become suspect. Third and more subtly, the primes suffuse all mathematics. The strands are engagingly interwoven by du Sautoy, evoking all the great mathematicians: Euclid, Gauss, Legendre, Cauchy, Hilbert, Fourier, Hardy, Littlewood, Ramanujan, Turing, and on, including of course, Riemann himself.

But what du Sautoy has written is more than a simple history of a single hypothesis; it is, in fact, a history of modern mathematics using prime numbers as a unifying theme. Buried in the story are flashes of insight and perception: that creativity can flourish under constraints; that while we believe mathematics to be universal and culturally neutral, every people overlays mathematics with its own cultural interpretations; that no matter how abstract and obtuse we believe an idea to be, sooner or later it is likely to be applied to practical situations. For example, if you make credit-card purchases through the internet, then you are probably aware that your credit-card information is encrypted via an algorithm based on prime numbers.

Another of du Sautoy's observations is that although mathematics has, in the past, been a search for patterns, more recently it has become a search for connections. Riemann's hypothesis has survived this shift in thinking, for although mathematicians have not verified the existence of a pattern in the sequence of primes, the search for such a pattern has given us a host of valuable connections between primes and other ideas. Many of those connections might not have been discovered without Riemann's hypothesis. In a real sense, this gets at the heart of "failure"—what does it mean "to fail" and how important is it? Should we expect mathematicians to be embarassed or ashamed because they have "failed" to prove Riemann's hypothesis? It is a point worth pondering because those failed attempts have, nevertheless, led to much new, enlightening, and valuable mathematics. Perhaps western cultures interpret failure too narrowly; perhaps we should place more value on what is actually accomplished rather than what is attempted; perhaps process is at least as important as product.

In any case, du Sautoy's book is a genuine achievement—rich at several levels, deft at handling technical details, humane in his treatment of personalities—a celebration of the human mind and spirit.

(jmh 15 June 06) © 2006 by J. M. Haile. All rights reserved.