Computers at home or in the office often sit idle for minutes, hours, or days at a time. The Internet now allows researchers to take advantage of this enormous reservoir of unused computer power.

More than 1.6 million people have downloaded software to sift through signals collected by the Arecibo radio telescope in Puerto Rico as part of a search for signs of intelligent extraterrestrial life. Investigators at the University of California, Berkeley manage the year-old project, known as SETI@home, and consolidate the results (SN: 9/18/99, p. 187).

A recent call for people interested in running large models of the world’s climate brought 15,000 replies within 2 weeks. “The response was pretty impressive,” says Myles R. Allen of the Rutherford Appleton Laboratory in Chilton, England. “It looks like we’re going ahead.”

Founded in 1996, a worldwide effort to identify record-breaking primes—numbers evenly divisible only by themselves and 1—has attracted more than 13,000 participants to the Great Internet Mersenne Prime Search (GIMPS) (SN: 2/21/98, p. 127). A smaller effort focuses on factoring large numbers to test encryption methods that safeguard information transmitted on the Internet.

Beyond the satisfaction of contributing to ongoing scientific and mathematical research, such participation can sometimes lead to a modicum of fame and even fortune.

Last summer, using GIMPS software on his home computer, technology consultant Nayan Hajratwala of Plymouth, Mich., discovered the first prime number with at least 1 million decimal digits.

As a result, he qualified for a prize of $50,000 offered by the Electronic Frontier Foundation in San Francisco, to spur development of cooperative computing technology. The foundation has since announced that it will award $100,000 to the discoverer of the first prime with at least 10 million digits.

“I still participate in the [GIMPS] program, and I am always looking for more computers to add to the search,” Hajratwala says.

Anyone can join in the hunt. All it takes is a desktop computer with idle time on its chips and a connection to the Internet. Indeed, the number of opportunities for taking part in group computational efforts continues to grow, and several Internet companies now focus on organizing and coordinating such projects.

“The combination of the Internet and the proliferation of home computers has resulted in one of the most significant new scientific and mathematical tools,” says retired engineer Harvey Dubner of Ridgewood, N.J., who keeps his own computers occupied with several prime pursuits. “Using normally wasted computer power to attack problems that cannot be solved in any other way is truly wonderful and important.”

**Central role**

A fascination with gargantuan numbers, especially those that are primes, has played a central role in the rise of collective computing. A decade ago, powerful supercomputers dominated the hunt for trophy-class prime numbers. Computer programmer George Woltman of Orlando, Fla., organized GIMPS to challenge that dominance.

The search for Mersenne primes, named for the French monk and mathematician Marin Mersenne, concerns numbers of the form 2^{n} – 1, where the exponent *n* is itself a prime. Written in binary notation, a Mersenne number consists of an unbroken string of 1s. Nearly all of the largest known primes are numbers of this form.

To facilitate searches for Mersenne primes, Woltman wrote a small, efficient computer program to test whether a number is a prime and made the software available on a Web site. His program relied on an ingenious computational algorithm invented by Richard E. Crandall of Reed College in Portland, Ore., to speed up certain computer operations. Anyone with even a modest personal computer could download and use the software to check lists of candidate numbers for primes. Because the computer is not connected to the Internet during the hours while the program runs, it is not vulnerable to unauthorized use by outsiders.

The project quickly attracted hundreds of participants. To manage the calculations, Scott Kurowski of Entropia.com (http://entropia.com/) in Campbell, Calif., developed the PrimeNet server, which now automatically doles out chunks of work and gathers results from thousands of copies of Woltman’s program residing on more than 22,000 computers throughout the world. Handling massive amounts of data processing over the Internet, the system represents a “new kind of computing service,” Kurowski says.

To date, GIMPS participants have found the four largest known primes. Mersenne primes are not the only targets of large group efforts. Last fall, Crandall, Ernst W. Mayer of Cupertino, Calif., and Jason S. Papadopoulos of the University of Maryland at College Park performed a massive calculation to prove that the 24th Fermat number, which is more than 5 million digits long, is not a prime.

That was the biggest computation ever done to obtain a simple “yes-or-no” answer, Crandall contends. It required 100 quadrillion (10^{17}) computer operations, comparable to the number needed to create the animated movie *A Bug’s Life*.

A Fermat number has the form 2^{2n} + 1. The first Fermat number is 2^{2} + 1, or 5; the second is 2^{4} + 1, or 17; and the third is 2^{8} + 1, or 257. Written in binary form, a Fermat number consists of 2^{n-1} zeroes sandwiched between an initial and a final 1.

**Not all prime**

In the early 1600s, French mathematician Pierre de Fermat conjectured that all such numbers are primes. That turns out not to be true. The first four Fermat numbers are prime, but the fifth is divisible by 641. Among the rest of the known Fermat numbers, up to and now including the 24th, none are prime.

Proving that a number is not a prime is much easier than actually determining its prime factors. No one has yet found even one prime factor of the 24th Fermat number, for example.

Crandall is now working with Kurowski to set up a system that would allow individuals and teams to join forces to factor large Fermat numbers. Crandall himself has offered modest cash prizes to anyone who finds new factors.

Esoteric pursuits of big numbers have value, Crandall notes. They have, for example, spurred the development of computational techniques useful for solving other problems. “Who knows where an idea will lead?” Crandall argues. Obscure notions can translate into tremendous benefits years later, he says.

**Time-consuming process**

Factoring has long intrigued and perplexed number theorists. Multiplication is easy, but the inverse process of finding the prime factors of a nonprime, or composite, number is horribly time-consuming when the numbers get large.

The assumed difficulty of factoring large numbers plays a crucial role in the so-called RSA encryption system, which is widely used to safeguard credit card numbers and other information transmitted across the Internet. To unscramble intercepted data, a snoop’s computer must factor a large number into its two prime-number components.

How big should the numbers be to ensure an adequate level of security? Since 1988, researchers have organized themselves into teams to test the security of these schemes (SN: 10/22/88, p. 263). Last year, one worldwide effort required 5 months on 300 personal computers and a Cray supercomputer to crack a 155-digit number into two 78-digit primes (SN: 10/2/99, p. 221).

Nowadays, information-security companies and other organizations occasionally offer cash prizes for either factoring numbers of a certain size or testing the security of alternative mathematical schemes for encrypting information.

The resulting competition has helped drive the development of increasingly efficient and refined methods for factoring, pitting one scheme against another. It has also produced strong rivalries. Last fall, for example, a global effort involving 195 volunteers and 740 computers demonstrated how a cryptographic system based on functions called elliptic-curve discrete logarithms could outperform an RSA scheme based on factoring.

The nonprofit Internet company known as distributed.net (http://www.distributed.net/) has coordinated several group attacks on cryptographic schemes, earning several cash awards. In January, its 62-day effort involving 38,000 contributors won a contest sponsored by a French security firm.

Most projects don’t offer prizes, however. The problems themselves are intriguing or have research value. The GIMPS Web site at http://www.mersenne.org/ maintains a list of mathematical research efforts soliciting support from individuals or teams.

**Deep questions**

Computational number theory isn’t the only field where individuals can contribute computing resources. “There are plenty of deep questions that could profit from extensive distributed computing,” says Joe P. Buhler of the Mathematical Sciences Research Institute in Berkeley, Calif., who has participated in several such efforts.

The SETI@home project takes advantage of idle computers to perform large-scale data processing. A colorful screensaver pops up whenever a computer is running the program, available at http://setiathome.ssl.berkeley.edu/.

Other opportunities span fields such as chemistry, molecular biology, epidemiology, population dynamics, and climate modeling. In the Oct. 14, 1999 *Nature*, Allen proposed recruiting people for the demanding task of forecasting climate. “For a 50-year forecast, we would first need to perform large numbers of simulations of the past 50 years, both with and without external influences such as increasing greenhouse gases,” he notes. “We could then discard all those perturbed models that were either inconsistent with the observed record or inconsistent with our (much less certain) estimate of what the 20th century would have been like in the absence of external influences.”

Allen calls the proposed effort Casino-21. “This experiment would introduce an entirely new form of climate prediction: a fuzzy prediction, reflecting the range of risks and probabilities, rather than a single ‘best-guess’ forecast,” he says. “We don’t have the computing resources to do this [fuzzy prediction] any other way.” There’s even the possibility of worldwide fame for the participant running the model that makes the most accurate forecast, Allen adds, “and a nice screensaver, of course.”

Participants can get into trouble, however, if they don’t own the computer that they’re using to run a climate model, check Mersenne numbers, or crunch radio-telescope data. In September 1998, computer consultant Aaron Blosser of Lakewood, Colo., found himself under arrest, accused of interfering with the operation of computers at the Denver-based US West phone company. With the permission of a supervisor, Blosser had installed GIMPS software on the company’s computers to search for Mersenne primes.

Soon after, computers at US West’s facility in Phoenix, Ariz., started to take as long as 5 minutes, rather than just 3 to 5 seconds, to retrieve telephone numbers. Company investigators discovered Blosser’s GIMPS program, blamed it for the slowdown, and called in the FBI, which searched Blosser’s home and confiscated his computers.

Blosser now faces the possibility of a misdemeanor charge of computer fraud and perhaps a fine and an indeterminate amount of restitution to US West for the cost of removing the software from any machine he had installed it on.

The lesson is clear. “Do not install your own software onto a work PC, or you could end up just like me,” Blosser warns. “If you do not own the machine you’re running the program on, you are in fact breaking the law if you don’t receive permission, in writing, from the owner of the machine.”

At the same time, Blosser adds, “I still believe in the validity of many of the ongoing research efforts, [especially] GIMPS. I can only hope that . . . companies and universities would be willing to explore the possibility of allowing their vast resources to be used in these projects.”