Motion System Design

In the loop: Prime time

Michael Cameron is a typical 20-year-old college student. He attends classes during the day and works in the evening. He also has a computer, but like many in his shoes, little time to use it. At the urging of a friend, Cameron decided to put his idle processor to work, digging around for prime numbers. The friend recommended a free program, Prime95, written by math enthusiast, George Woltman. On November 14, 2001, after running for 42 days, Cameron’s PC produced an extremely rare number, a Mersenne prime, etching his and Woltman’s names into the annals of mathematical history.

The quest for these magical numbers stretches back more than 2,000 years to the master mathematician, Euclid himself. Best known for his pioneering work in geometry, Euclid was also one of the first to discover the unique properties of prime numbers. In 300 BC, he initiated a search for those of a special form called Mersenne primes.

Any integer greater than one is considered prime if it is indivisible except by one and itself. Thus, the first few prime numbers are 2, 3, 5, 7, and 11. Mersenne primes — named after Marin Mersenne, a French monk — are a subset of ordinary primes. Following the form 2n-1, they include 3 (n=2), 7 (n=3), 31 (n=5), 127 (n=7), and so on to infinity. Over the years, many notable mathematicians have sought after and discovered these most primal of primes, including Euclid, Descartes, Leibniz, Euler, Fermat, and Landry.

Cameron’s find, 213,466,917-1, is only the 39th in the Mersenne series. If written out, its 4,053,946 digits would fill a book. In a sense, this gargantuan figure is now Cameron’s number. Although it may have no particular significance, who’s to say it won’t show up in a physics book someday, alongside Avagadro’s number, Planck and Boltzmann’s constants, and the speed of light. Perhaps it is a key multiple of some galactic proportion, or an inverse factor of some yet undiscovered elementary particle. Only time will tell.

In the meantime, the not-so-trivial pursuit for Mersennes and other such numbers will continue. Along the way, as we face increasingly complex obstacles, we will undoubtedly run into more surprises. Primes inadvertently skipped over may be exposed. New information may arise, hidden not in the numbers themselves perhaps, but in their distribution. It was in the early stages of this very quest that we gained some of the greatest insights in elementary number theory. With computers now in the hunt, who knows what incredible discoveries lie ahead.

Competition to be first, to stake a claim in history, will further accelerate the development of computer technology. Indeed, many multiplication algorithms used today owe their existence to the study of large prime numbers. In 1968, for example, someone pursuing primes perfected Fast Fourier Transforms, now widely used in — drum roll, please — motion control.

Another byproduct of the numeric space race is computer tests. Since the dawn of the digital age, programs for finding primes have been used to put new, unproven hardware through its paces. Software routines like those employed by Cameron were, in fact, used by Intel to test Pentium II and Pentium Pro processors prior to shipping. The infamous Pentium bug — I still own one of those flawed 90- MHz chips — was originally found in an attempt to calculate the twin prime constant. Even supercomputers are debugged this way. Cray Research, for example, tests hardware with a prime number checker developed by one of its employees, a guy who has helped find more Mersenne primes than anyone else.

As for Cameron’s Mersenne, there is yet another reward — a $5,000 check that he can spend on anything he wants. But there’s more. An even bigger prize awaits the person who discovers the next Mersenne. The amount, though no prime number itself, is certainly “prime time” at a cool $50,000; and it could be yours.

For details on how to bag the prize, log on to our Web site and go into our livelink section, Direct Connect. There you will find a familiar number, 213,466,917-1, and by clicking on it, you’ll be transported to all the information you need, in addition to the free search routine. Good luck, and don’t forget us if you win.

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