INCREASINGLY SOPHISTICATED apparatus and ever-more powerful computers are finding their way into virtually every branch of science. But in one field of research the only essential equipment is a pen, some scraps of paper and a comfy chair.

Mathematics underpins virtually every other branch of the pure and applied sciences. However, in its most theoretical form, maths is entirely separate from other disciplines as its exists only within the mind.

Indeed, Dr Marcus du Sautoy, a Royal Society research fellow in mathematics at the University of Cambridge says he rarely uses a computer - except as a word processor to write up his findings. He is used to dealing with non-mathematicians' bafflement over how he spends his time - how can someone produce useful work simply by sitting and thinking?

"Yes, most people seem to think that mathematical research is long division to a lot of decimal places. That's because of the maths they remember from school," he says. "At that stage you are just learning the basic techniques, the language and grammar of mathematics. There are similarities with music; first you have to learn the scales and time signatures but at least with music you do have an idea what it will eventually sound like. You don't really learn to `hear' a piece of mathematics until you get to university level."

Du Sautoy's own area is symmetry, an enduring obsession of mathematicians because the harmonious patterns they describe on the page keep cropping up in the real world - for example, the most efficient way of stacking oranges in a greengrocer's box forms a regular hexagonal shape.

There is also the hope that his formulae will help explain other mathematical phenomena such as prime numbers. "The thrill of maths is in spotting and finding patterns which exist in two different areas. It's like wandering around a darkened building and using a mathematical torch to try to find the passageway connecting two different rooms."

Prime numbers fascinate Du Sautoy and his colleagues because they are the fundamental particles of their world - the atoms of the mathematician's Periodic Table. The search continues for ever higher prime numbers and to explain why they occur so randomly. To explain their distribution would answer one of the great mysteries of maths, the Riemann hypothesis.

A trick using prime numbers was invented by the great French mathematician Pierre de Fermat more than 250 years ago. This now forms the basis of the encryption system used to prevent eavesdropping on electronic communications. This prevents, for example, somebody intercepting an e-mail giving a person's credit card numbers. Indeed, the security of the whole financial system is dependent on relatively simple codes using prime numbers relayed between the sender and authorised receiver of the message (or rather their computers). The actual prime numbers used are only known by the authorised receiver; in fact, the sender cannot even decode his own message once it has been encrypted.

Such unexpected applications of mathematical knowledge are not uncommon. "We do what we are doing because of the pleasure we get from understanding the mathematical world. Mostly the problems we set out to solve do not relate to the real world but you never know when they will - sometimes you do get these bizarre pay-offs."