Text: Prime numbers not so random? A kind of order may be buried in the occurrence of indivisible numbers. 24 March 2003 PHILIP BALL A team of physicists may have stumbled upon a surprising discovery about one of the deepest and best-studied questions in pure mathematics: whether or not prime numbers appear randomly in the sequence of whole numbers. Primes cannot be divided by any smaller whole number other than 1. Pradeep Kumar and colleagues at Boston University1 reckon that they have found a kind of order among the distribution of primes, the numbers that cannot be divided by any smaller number other than 1. The first few primes are 2, 3, 5, 7, 11 and 13; the largest currently known has over 4 million digits. No one has yet proved that their occurrence follows any pattern, or whether there is definitely no pattern. Kumar's team looked at the increments in the intervals between consecutive primes. For example, the intervals between the first few are 1, 2, 2, 4 and 2. The increments are the differences between these successive intervals: +1, 0, +2 and -2. These increments are not random, the physicists conclude: they have a rough-and-ready predictability. "Positive values are almost every time followed by corresponding negative values," explains team member Plamen Ivanov. That is clearly already true for the third and fourth increments above: +2 and -2. The researchers are not experts in number theory, the relevant branch of pure mathematics. In fact, they did not set out to study the statistics of prime numbers at all. Ivanov suggested that his graduate student Kumar use primes merely to dry-run a statistical tool that they had developed to study heartbeat rhythms. While probing the variations of the gaps between heartbeats, the researchers found something else. A plot of the number of increments of different sizes shows oscillations with a period of three. That is to say, increments of plus or minus 6, 12, 18, and so on, are statistically less likely than increments of other sizes. Excepting the first in the series, the increments are even numbers, as all primes other than 2 are odd. That's why this oscillation has a period of 3 rather than 6, as it appears to have. This finding is less surprising. Previous studies found period-6 oscillations in the histogram of distances between consecutive primes. Increments, remember, are differences between consecutive distances. The Boston team's findings are not supported by any kind of rigorous mathematical proof. So sadly they can't shed any light on one of the biggest problems in maths: the Riemann hypothesis. This conjecture in number theory is intimately related to the distribution of primes. In 2001 the Clay Institute in the USA offered a prize of a million dollars for a proof of the Riemann hypothesis. But The findings might have implications in the real world, as some systems in physics and biology - such as interacting prey and predator species with different life cycles - show patterns that depend on prime numbers. References # Kumar, P. Ivanov, P. C. Stanley, H. E. Information entropy and correlations in prime numbers. Preprint. http://xxx.lanl.gov/abs/cond-mat/0303110 (2003). |Article| © Nature News Service / Macmillan Magazines Ltd 2003

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