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.

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".

They've done it again! GIMPS has discovered the largest known prime number: 2^57,885,161-1. This massive 17,425,170 digit number was discovered thanks to clever distributed computing software that uses idle computer time donated by volunteers.

The prime numbers are the atoms amongst the integers, and while we know that there are infinitely many of them, there's no general formula that generates them all. Julian Havil looks at a little-known algorithm that sieves out all primes up to a given number, and which is astonishing in its simplicity.

The primes are the building blocks of our number system, but there's no general formula that will give you all of them. If you want them, you have to hunt them down one by one. Abigail Kirk investigates a method that does just that.

The first third degree transcendental L-function

The Great Internet Mersenne Prime Search strikes again

The GIMPS project finds a new Mersenne prime

A major advance towards the twin prime conjecture

7.8 million digits down, only 2.2 million more needed!