I always feel a mixture of relish and dread when my neighbour at a party asks: "So what do you do?" Finding out that I am a mathematician, most people think that I must sit in my office all day doing long division to lots of decimal places. But that misses the true essence of what it means to be a mathematician.

Above all else, for me, a mathematician is a pattern searcher. Maths is about finding patterns in the chaos of numbers that surrounds us – to find the music which binds all these numbers together. And of all the numbers in mathematics, it’s the primes which offer the greatest challenge of all to the pattern searcher.

As you count through these indivisible numbers it is extremely difficult to predict where you’ll find the next prime number. Like the stars in the night sky, the primes appear scattered randomly through the universe of numbers. Some are clustered close together – others are flung far apart. There just doesn’t seem to be any pattern. It doesn’t make sense. And if there’s one thing a mathematician craves, it’s pattern and sense.

This problem of the pattern - or lack of pattern - of the primes has been like a magnet to perplexed mathematicians ever since the Ancient Greeks first proved they go on for ever. But it has outwitted the greatest mathematical minds of the last two thousand years.

It wasn’t until the late 18th century that a breakthrough finally happened. It was made by a 15 year-old German boy, who grew up to be one of the greatest mathematicians who ever lived: Carl Friedrich Gauss.

It was a present he was given for his 15th birthday that was to change the course of mathematical history. The present was a book of mathematical tables. At the back of the book was a list which began to obsess the young Gauss: a table of prime numbers. He spent hours pouring over these tables, trying to force them to reveal their secrets. His efforts eventually lead to an amazing discovery.

Faced with this table of unpredictable primes, Gauss executed one of the classic moves in the mathematician’s arsenal: if things get too complicated, do some lateral thinking. Look at the problem in a new way. Ask a new question. Instead of attempting to predict precisely which numbers are prime, Gauss asked instead how many primes there are.

Gauss started counting how many primes there were up to 10, up to 100, up to 1000, and so on. As he counted higher and higher, the primes seemed to be getting rarer and rarer. But was there some way to predict how they thinned out? As he counted his primes he realised he could calculate the probability of getting a prime. For example, there are 25 primes up to 100. That means there is a 1 in 4 chance that a number between 1 and 100 was prime. But between 1 and 1000, there is only a 1 in 6 chance of a prime occurring. Perhaps Nature chose the primes using a set of prime number dice.

But could Gauss predict the number of sides on the dice as Nature chose bigger and bigger primes? As he counted higher and higher, working out the prime probability, Gauss began to see a pattern emerging. Despite the randomness of the primes, a stunning regularity seemed to be looming out of the mist.

As he added a nought, Gauss realised that the proportion of prime numbers was decreasing by the same amount every time – by about 2. So from 10,000, to 100,000, to 1,000,000, the chance of getting a prime decreases from 1 in 8, to 1 in 10 down to 1 in 12. It was as if the primes around 10,000 were chosen using an 8 sided dice whilst those around 1,000,000 were chosen using a 12 sided dice.

Using his guess at how the primes thinned out, Gauss was able to estimate roughly how many primes you would expect to encounter the further you counted through the universe of numbers. For example, in the numbers from 1,000,000 to 1,000,120 the 12 sided prime number die would predict 10 primes in this region. But it was not an exact formula. And if it’s one thing mathematicians crave it’s precision.

    next > Page 1 of 2

Content last updated: 12/09/2005

Comments