## number theory

*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?

**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.

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

**Yutaka Nishiyama**explains why, and how beautiful mathematical oddities can inspire us to discover new mathematics.