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Divisible only by themselves and the number one, prime numbers underlie the security of international banking. They also represent one of the most tantalising enigmas in the pursuit of human knowledge. For centuries mathematicians have striven to find a way of predicting when the next prime number will occur in a list of numbers. Is there a formula which could generate primes? Why does it matter? Tonight Marcus de Sautoy goes in search of mathematics' Holy Grail - and explains why so many mathematicians have gone mad on the same journey. Marcus du Sautoy is professor of mathematics at Oxford and Royal Society University Research fellow. In addition to scientific papers, he has written and broadcast widely and is author of The Music of the Primes. In the chair is Daniel Glaser, neuroscientist at UCL.