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Music of the Primes by Marcus du Sautoy Perennial/Harper Collins $14

"Prime numbers by a Frenchman - more mathematics ... only a geek could like this book". That was the reaction I got at the Easter dinner table when I tried to convince members of the family to take a look at Marcus du Sautoy's excursion through the current world of mathematics. But like a bug, okay even a virus, Marcus's Music of the Primes plays on some interesting ideas and theme notes - and suddenly, if you have any strain of intellectual curiosity, suddenly you find this book is infectuously fascinating.

First the mathematics of primes has pedigree. Going back to the Greeks and Euclid in 300BC at Alexandia Egypt, the primes were recognized as special numbers - the ones that had no divisors - other than themselves. These are the atoms and primary elements of the numberline and mathematics. And Euclid, famous for his Elements and the 500 hundred theorems and proofs that established geometry and engineering, he is the one "who had no idea how to produce primes explicitly, but he could prove that they never ran dry". Primes were infinite. "...gone was the chance of fitting together a Periodic Table listing all of the primes".

Next the mathematics of the infinite primes has a solution, but a brute force solution. Another Greek, Erasthones, showed with his Sieve method how to enumerate the primes up to any arbitrary number, N. First strike out all numbers up to N that are multiples of 2, then strike out from the remaining numbers all those that are are multiples of the next prime, 3. Continue through all the primes upto (N+1)/2. If N has not been struck out, then it is prime. Add it to the list of Primes. This method is first order timesaving because it eliminates the need to find out if a number is not prime more than once. Nonetheless it is far from being efficient and that turns out to be a virtue.

The insoluable mathematics of producing near infinite primes is critical for computer and information security system. In fact, at data security's core is RSA. And RSA is a coding scheme that depends on using huge primes. As du Sautoy notes "the extraordinary thing is that although the construction of this [RSA] code depends on [the efficiency of] discoveries made by Fermat over three hundred years ago, to break this code code depends on a problem we still can't answer. The security of RSA depends on our inability to answer basic questions about prime numbers."

So part of the Music of the Primes is their crucial value today in all busines and computer transactions. Primes provide the nearly unlockable keys to security codes. But these keys are not totally unlockable. A contest was sponsored by RSA owners open to anyone in the world to unravel an RSA-coded message using primes. The code was deciphered correctly after months of work, harnessing hundreds of computers. Brute force will break the basic RSA code; but the locksmiths are a step ahead of the codebreakers with variants and ever larger primes.

However, if you read Music of the Primes for insights into data security - its like reading a Julia Child cookbook for fastfood preparation. The du Sautoy sauce is the ability to make the work and lives of mathematicians so compelling and interesting. And as du Sautoy reveals mathematicians are far from being geeks or idiots savants. In fact they are pattern seers and discoverers in worlds of finely tuned logic.

As du Sautoy describes the world of primes one becomes acquainted with the brash and brilliant Fermat, the religous fervor of Mersenne, and the astonishing cleverness of Gauss - "Indeed, Gauss's schoolteacher liked to set this problem of adding up the first 100 numbers for his class, knowing that it always tooks his students so long he could sneak in forty winks.... While the other students began labouring away, within seconds the ten year-old Gauss had laid his tablet on the table. Furious, the school teacher thought that Gauss was being cheeky and must have cheated somehow. But the pupil explained that all you needed was to put N= 100 into the formula 1/2 * (N+1) * N = 5050 and you will get the sum to 100 right away". Gauss had seen or discovered the formula in the summing of the numbers.

And in a similar fashion readers visit with Euler, Riemann, Hardy,Turing among other Prime mathematicians. And each has a story to tell. In fact, du Sautoy is a mster narrator: quickly, yet skillfully sketching out a personality portrait while also unraveling a fascinating history lesson about the culture and mores of the era . In effect we get a searing glimpse into the status of scholarship, learning and sheer human civility in 18th century Switzerland, 19th century Prussia, British Empire England, and all the pressures and consequences of World War II secret service work.

Along the way we dance around the tantalizing secrets of the Riemann Hypothesis. See its importance as defined by Hilbert at the turn into the 2oth century, and see all the attempts by towering minds to establish its veracity. Even aided by mechanical idiots savants in the guise of ever more powerful computers - the Hypothesis still stands unproven 105 years later.

And so du Sautoy proves once again that it is the nature of the journey as much as the end point that is inherent in fascination, interest, innovation and discovery. Along the way, the math and logic has its own muse and music which du Sautoy uncovers for us in confident revelations - its is like taking a Jay Gould walk through evolution or a Carl Sagan tour of the universe - remarkably informative and entertaining. And don't miss the free wonders of du Sautoy's Music of the Primes website.

(c)JBSurveyer 2005