number theory

 Number theory is famous for problems that everyone can understand and that are easy to express, but that are fiendishly difficult to prove. Here are some of our favourites. Agreeing to pay £50,000 for something worth £2 wouldn't win you any haggling competitions. In mathematics, however, a similar result can bring you international acclaim. This is the case with recent progress towards the famous twin prime conjecture. An "electric atomosphere" is not what you expect at a maths lecture. But it is what prevailed when Andrew Wiles announced his proof of a 350-year-old-old problem, Fermat's last theorem, exactly 20 years ago. This year's Abel Prize has been awarded to the Belgian mathematician Pierre Deligne for "seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields". This year's Abel Prize goes to Endre Szemerédi for his "fundamental contributions to discrete mathematics and theoretical computer science." 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. Results in mathematics come in several flavours — 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.