THE PURSUIT of the largest known prime numbers is an occasionally eccentric endeavour that involves tens of thousands of researchers and scientists around the globe. Now UCT's Dr Ken Hughes has entered the mathematical mÃªlÃ©e after serendipitously happening upon a possibly unique formula for generating prime numbers of quite respectable size.
It all started in March 2002, explains Hughes, when he was setting his first class test for his third-year algebra class. "Usually testing students involves setting routine problems of no great originality or profundity, and one does not anticipate getting any research dividend out of it," he reflects.
This time around, however, he set a question about algebraically deriving the properties of the Mersenne numbers – named after priest-cum-mathematician and friend of Descartes, Marin Mersenne (1588–1648). Here the law of formation of the series is to double and add one, thus 1, 3, 7, 15, 31, 63... Many Mersenne numbers also happen to be prime numbers, i.e. numbers that lack factors other than themselves and one, explains Hughes. All the biggest known prime numbers today are Mersenne primes, with the largest, known as the 38th Mersenne prime, all of 2 098 960 digits long, and a 39th Mersenne prime – boasting more than four million digits – currently being verified (see www.utm.edu/research/primes/largest.html
It was while setting the paper that Hughes wondered whether the algebraic method he was using could be applied to the numbers uncovered by Mersenne's correspondent, the French mathematician Pierre de Fermat (1601–1665), who also happened to be the founder of number theory. Fermat numbers are 3, 5, 17, 257 . . . and their law of formation is a little more complicated, notes Hughes, having to subtract one, then square and add one. "Fermat had this idea that there were families of primes other than Mersenne primes, and there would be an infinite number of them," he adds.
Fermat, who had to do without the convenience and speed of a personal computer, calculated his numbers the hard way (by hand) and erroneously figured that all first six of his numbers were prime. Nigh on 150 years later it was proven, however, that while the first five were primes, the sixth, alas, was not (none of the next 20 numbers after the first five are).
Hughes discovered, to his delight, that his little piece of algebra happened to produce a whole new family of generalised Fermat numbers – 7, 73, 32257... – with a law of formation involving cubes. "Amazingly, my method applies easily to generate this series, and no-one else seems to have noticed this over the last 300 years," he says.
"Furthermore, the method generalises readily and produces a whole series of families of generalised Fermat numbers." While Hughes is still uncertain whether his method of deriving Fermat numbers is unprecedented, so far his research has unearthed no other claimants.
With the Fermat numbers growing rather rapidly, Hughes realised the need for computation – he, too, had calculated the first few by hand – and help came (again fortuitously) in the shape of first-year student, "the amazing Thomas Ludwig".
Ludwig, who approached Hughes on some unrelated mathematical matter, happened to have a programme to test primality on his powerful PC at home, and volunteered to redirect his computer time to help check Hughes' findings.
"In the interest of science, he has factored some truly awesome numbers of 30 digits or more," reports Hughes.
Should the numbers grow much bigger and too much for Ludwig's computer, there may be a need to involve others in the pursuit.
And with prime numbers having a general interest, Hughes has now opened his seminar to undergraduates.
Once the checking is done, he will find a suitable forum to publish his research, and would very likely do so via the web, Hughes indicated. "We believe it should be possible to generate some very large Fermat numbers to compete with the Mersenne numbers.
"We're not so much trying to come up with the world's largest prime, but simply to come up with an alternative to the Mersenne primes."
Prime numbers have an element of superstition and mystery about them, which may explain their popularity, Hughes notes. In the United States, about 120 000 people form part of a computer conglomerate – sharing computer time – known as the Great Internet Mersenne Prime Search, or GIMPS, which every few months or years pops up a new prime number (such as the 38th and 39th Mersenne primes).
"The basic structure of mathematics and physics turns around numbers," Hughes says, "especially prime numbers.
"One's ultimate hope when doing number theory is to throw light on the nature of reality."