Prime secret only an atom can unlock

Euclid Cartoon

Mathematicians are notorious for using jargon recognised only by other mathematicians.There is,however,one term they bandy about whose aptness cannot be gainsaid : "prime" numbers.

With every whole number greater than one being itself prime (not divisible by any other number other than one and itself) or made out of them,prime numbers bear all the hallmarks of coming straight from the Great Mathematician in the sky -pure,essential,inexhaustible.

How odd,then,that more than 2,000 years after the Greek mathematician Euclid first proved that there is an infinite number of them,no one has found a formula that spits out more than a few hundred of the primes 2,3,5,7,11 and so on.

The challenge of finding such a formula has attracted amateur and professional mathematicians alike.One of the simplest and earliest was discovered by the great 18th century Swiss mathematician Leonhard Euler,who came up with the following recipe which produces 40 primes: for each  of the numbers 0,1,2 and so on,take the number,add it to its square and add the result to 41.The result is a selection of primes from 41 up to 1601.

The advent of computers has led to the discovery of more complex formulae for capturing capable of generating more primes.One found in 1989,involved cubes as squares,generates an impressive 267 primes - though still,of course,some way short of the infinitude of primes.

This lack of a formula is made all the more frustrating for the knowledge that there is,in fact,such a recipe out there - if only we could work out how to use it..In 1971 the Soviet mathematician Yuri Matijasevic proved that there was a formula that generated all the primes,and it involved powers as high as 37 -although  he did not actually write the formulae down.

Five years later a group of mathematicians led by the American James Jones not only found a less unwieldy formula,involving merely 25th powers,but wrote it down as well.The bad news,however,is that - like every similar formula ever found the Jones recipe contains  a set of symbols whose numerical value is unknown.Without these,the recipe is quite useless - though of course it remains a major theoretical achievement.

This frustrating state of affairs suggests that perhaps we are looking in the wrong place for the secrets of the prime numbers.An ingenious bit of lateral thinking by an Italian physicist now suggests that it might be worth exploring somewhere quite unexpected: inside the atom world.

From the very earliest days of quantum theory it was it was clear that mathematics ruled the roost in the sub-atomic world.Over a century ago the Swiss amateur physicist Johann Balmer pointed out that the energy levels of hydrogen - as revealed by the frequencies of its spectral lines - followed a simple mathematical rule involving the reciprocals of the squares of whole numbers.

Balmer's discovery,made after countless hours of trying to make sense of experimental data,was later shown to be the result of the way in which the nucleus of the hydrogen atom clings to its single orbiting  electron.

Giuseppe Mussardo,a theoretical physicist at the International School for Advanced Studies in Trieste, Italy,has now shown that it may be possible to put together a collection of atoms whose spectral lines are in sequences other than inverse-squares.And one of these sequences is in prime numbers.

The details are rather technical,but what makes Mussardo's idea so clever is that it does  not require a recipe for generating primes.Instead, it relies on a formula showing how widespread primes are among among the ordinary numbers.

Computer searches have shown that primes become rarer as they get bigger.While 25 per cent of the numbers are below 100 are prime,only 5 per cent below a billion are - and mathematicians have established a number of recipes underpinning this decline.

Mussardo has taken one of these recipes  and used it to reveal some details about how an atom must cling on to its electrons if its spectral line frequencies are to mimic the sequence of primes.

All that remains now is to find a suitable atom.Whether it exists is as yet unknown;almost certainly it will have to be artificially concocted,perhaps using groups of atoms,which may wreck the phenomena Mussardo is trying to exploit.But can the Great Mathematician really refuse so ingenious an assault on so frustrating a problem?

Robert Matthews

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