## number theory

The number 1 can be written as a sum of unit fractions, that is fractions with 1 in the numerator. But how long can we make such a sum?

One of the most surprising things about mathematics is its many unsolved mysteries. Mathematics is far from "done and dusted", and Steve Humble shows us how we can come up with some mathematical mysteries of our own.

*theorems*are the big important results,

*conjectures*will be important results one day when they are proved, and

*lemmas*are small results that are just stepping stones on the way to the big stuff. Right? Then why has the Fields medal just been awarded to Ngô Bào Châu for his proof of a lemma?

Eron Lindenstrauss got the Fields Medal for developing tools in the area of dynamical systems and using them to crack hard problems in the seemingly unrelated area of number theory.

On March 14 2010 a mathematician and a magician teamed up to perform what they believed to be the world's largest live magic trick. The trick involved a thousand volunteers from the around the world who, using free choice, each came up with a number that was only known to themselves. And although the volunteer might be on the other side of the globe, the mathematician and the magician were able to read their mind and tell them which number they had chosen.

**Abigail Kirk**investigates a method that does just that.

**Neil Pieprzak**tells the fascinating story of Andrew Wiles who, with intense devotion and in secret, proved a deceptively simple-looking conjecture that had defeated mathematicians for almost 400 years.