Thanks to the recent film *A Beautiful Mind*, and a book of the same name on which it was based, John F Nash is one of the best known mathematicians of our age.

His story is fascinating. In the late 1940s, at the age of just 21, John Nash revolutionised our understanding of game theory, an area of research which seeks to explain how individuals, corporations or nations compete against each other.

Half a century later he was short-listed for the Nobel Prize in Economics, which resulted in a controversy so great that it almost tore apart the committee responsible for the award.

If you wonder why it was in Economics rather than mathematics, you need to know that nobody receives a Nobel prize in mathematics - they are not awarded.

The work which gained him this honour some 50 years later, was written in mathematical terms, but was based on ideas central to economics.

However, it was not the significance of his breakthrough that was under debate, but rather his mental state. For three decades, Nash had suffered from severe paranoid schizophrenia, which had cost him his job, his family, and his freedom…

John Nash was born in a small, remote American city called Bluefield in the Appalachian mountains. As a child he found it hard to make friends, preferring instead to play by himself and undertake scientific experiments at home. One of his chemistry experiments at school led to the death of another student, and his school friends taunted and bullied him because of his behaviour.

He obtained a full scholarship to Carnegie Mellon University in Pittsburgh and specialised in chemical engineering. However, he was quickly frustrated with chemistry and switched to mathematics, and was eventually awarded a Masters degree in addition to his Bachelors because his work was so advanced!

It was at Carnegie that he first came across economics, which eventually led to his famous paper The Bargaining Problem and sparked his interest in game theory.

He continued to Princeton, where he was given a fellowship, although he did not attend many lectures, using his time to develop his own highly original approach to mathematics. During this time Nash invented a topological game named after him, which was similar to Hex invented by a Dane called Piet Hein. Nash also spent time discussing his ideas on gravity with Einstein (also at Princeton), who advised him to study more physics!

After he was awarded his doctorate for his thesis on Non-Cooperative Games in 1950, he went to work for the RAND Corporation, where he applied his experience in game theory, to military and diplomatic strategy on the Cold War Conflict. He was dismissed after he was arrested in a police sting operation.

He went to the Massachusetts Institute of Technology (MIT) in 1951 as an instructor, where he stayed until 1959.

The year 1958 was a particularly difficult time for Nash. His wife was pregnant, and he began to exhibit symptoms of schizophrenia - a severe mental illness. He believed that the New York Times contained specially encoded messages from outer space addressed to him. He resigned his post at MIT, and was hospitalised for 50 days of his own free will, although on his release he went to Europe where he hoped to gain refugee status. This was not successful, and on returning to the States his condition deteriorated and the next years were marked by periods of extreme delusion and hospitalisation interspersed with periods of rationality. During his lucid moments he continued to produce mathematical papers and research, and over a long period eventually began to recover.

He was awarded the Nobel Prize in Economic Science in 1994 (jointly with Harsanyi and Selten) for his work on game theory.

While he was still at school, Nash began to read advanced mathematics, and inspired by a book by E T Bell called *Men of mathematics*, set about proving Fermat’s Theorem. No, not Fermat’s Last Theorem, but a result called by mathematicians, Fermat’s Little Theorem. It may be written like this. Suppose that *a* is a whole number and *p* a prime number, then

a^{p}-a\ \mbox{is divisible by}\ p

That he managed to prove it was an incredible achievement for someone so young. As an example of its use, we know that

a^{17}-a

where a is any whole number, is **always** divisible by 17. So:

\qquad 1^{17}-1\ \mbox{is divisible by 17;}

\qquad 2^{17}-1\ \mbox{is divisible by 17;}

\qquad 3^{17}-1\ \mbox{is divisible by 17;}