A MATHS hobbyist has beaten an international computer network in a race to test the Riemann hypothesis, an important mathematical conjecture about prime numbers that has remained unproven for almost 150 years.
The hypothesis says that the prime numbers are distributed randomly along the number line, rather than following any pattern. But although it was put forward in 1859 and now underpins many other theories, despite a $1 million prize offered by the Clay Mathematics Institute in Cambridge, Massachusetts, no one has yet been able to prove it.
Instead of trying to prove the hypothesis, Stephen Wedeniwski, a computer scientist with IBM in Germany, started the ZetaGrid project () in 2001 to hunt for examples that would prove it false. That鈥檚 possible because the hypothesis can be expressed as a function (see Graphic) whose output value oscillates around zero as the input changes.
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If the hypothesis is true, the value hits zero only when the input is a number of a particular form. The discovery of a single zero produced by a different form of input would be enough to bring down the conjecture.
The ZetaGrid divides the job into small parts that can be checked using the spare processing power of computers belonging to volunteers, a so-called distributed computing project. Wedeniwski had recruited 11,000 machines and was creeping towards the milestone of 1 trillion zeros when Xavier Gourdon, an amateur mathematician who works for an engineering company near Paris, stole the group鈥檚 thunder.
Gourdon has now announced that he has checked 10 trillion zeros using only 25 computers. 鈥淭he first surprise was that he had calculated so many zeros. The second surprise was that his algorithm was so good,鈥� says Wedeniwski.
While Wedeniwski鈥檚 program computes the zeros one at a time, Gourdon鈥檚 algorithm locates many in one go. This makes it 10,000 times as fast.
The 10 trillion zeros checked so far all satisfy the hypothesis. That will reassure mathematicians, but it does not prove the hypothesis because a stray zero could still lurk undetected. But neither Gourdon nor Wedeniwski think that will happen. So why do they bother?
鈥淲hy do people climb mountains?鈥� retorts Andrew Odlyzko, a mathematician at the University of Minnesota in Minneapolis. Gourdon鈥檚 algorithm extends an approach that Odlyzko invented in the 1970s, and shows that computing power isn鈥檛 necessarily the most important factor when it comes to tackling problems like this, says Odlyzko. 鈥淧rogress in algorithms is in many cases of comparable importance,鈥� he says.
Gourdon has now offered his algorithm to the ZetaGrid project. 鈥淚 always planned to give them my program, but first I wanted to prove it was good.鈥�
