Review of Prime Obsession

By Eric Wolff

In educated society, it is unacceptable to forget the date of the Declaration of Independence, or to not know that William Shakespeare was a playwright. But, for some reason, it’s acceptable to claim ignorance of grade school arithmetic. Consider, as an example, the tolerant chuckle given to a diner unwilling to work out the party’s tip because he was "never any good at math."

Despite the general acceptance of, even pride in, our incompetence at this basic skill, we must be thankful that there are men and women in our midst who spend their days pondering the mysteries of numbers and working to create those most definite of all truths, mathematical proofs.

Occasionally an individual appears, a visionary, who can see in the numbers broad patterns that are not proven, but still probably true. Pierre de Fermat was one of these, and his Last Theorem (there are no whole-number solutions to the equation x ⁿ +y ⁿ =z ⁿ  for  any whole n>2) became one of the most famous unsolved problems in mathematics — until Andrew Wiles did solve it, in 1994.

Riemann’s Hypothesis is both more difficult to express and vastly more useful (Carl Gauss, arguably the greatest mathematician of all time, felt that Fermat’s Last Theorem wasn’t worth his effort), and mathematicians have spent the last 150 years delving into its mysteries.

The RH, as mathematicians know it, has applications beyond its native number theory, and pops up in geometry, field theory, even quantum physics.

Understanding the RH normally requires several college-level math courses, but, fortunately, two recently published books both purport to explain it and its importance.

In a perfect world, the publishers would have stamped a great big "DON’T PANIC" across Prime Obsession and The Riemann Hypothesis.  They are not scary books. They are aimed at the likes of you and me, the math uninitiated; I, personally, haven’t seen the inside of a math class since high school, but found both books entirely comprehensible.

At its most basic level, the Riemann Hypothesis proposes a means to calculate, with perfect accuracy, the number of prime numbers — numbers only divisible by themselves and one —less than any given other number. In fact, the title of Riemann’s paper to the Berlin Academy in 1859, the one that proposed the hypothesis, was "On the Number of Prime Numbers Below a Given Quantity."

This is no lightweight question, nor should it be written off as a mental game for the very smart. These days, prime numbers are integral to digital audio quality — i.e., CDs and mp3s — and they are the basis for the encryption that keeps credit-card numbers safe when they’re sent through the Internet. Connections to the hypothesis have appeared in such unexpected quarters as the behavior of electrons in an atom.

To explain the Riemann Hypothesis, or RH as mathematicians call it, John Derbyshire staggers the chapters of Prime Obsession — even-numbered chapters are history, biography, and anecdote; odd-numbered are math and proof. The two threads nearly stand alone as separate works and don’t become entangled until the end.

Mr. Derbyshire’s tone is warm and witty, and, reading his book, I felt as though he was sitting next to me, guiding my ascent into one of math’s more rarefied fields. He puts to good use his own training in higher mathematics and a wealth of interviews and research to carefully articulate both the history and the mathematics of RH. The historical portions of the book provide much-needed rest from abstraction, and Mr. Derbyshire takes the time to introduce mathematicians both legendary, like Carl Gauss, and contemporary, like Atle Selberg. He takes particular pains to sketch the life of Georg Bernhard Riemann himself,the quiet scholar who died of tuberculosis at the age of 39, and the man whose spirit Mr. Derbyshire imagined "moving around discreetly behind the scenes in both my mathematical and historical chapters."

Prime Obsession divides further, into pre- and post-1900 sections. The first allowed me to feel smart for my ability to understand the work of these geniuses. The second forced me into some serious mental labor, and, for the first time, Mr. Derbyshire reduces components of his explanation — some of which require a semester of college math — into "trust me, this works." I can’t blame him for this; only a great talent for explanation allowed him to get me this far.

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Many people have terror on sight when it comes to the arcane symbols of mathematics, and they might not consider picking up a book about a problem the greatest minds in the world haven’t solved.  That would be a shame; like any great teacher, John Derbyshire’s passion for his subject transforms unfamiliar and difficult material into a genuinely enjoyable and enlightening experience. Isn’t that why we read?