Here's the ancient proof that there are infinitely many prime numbers.

James Maynard, one of the prize winners at the European Congress of Mathematics, is counting primes that don't have 7s in them. But why?

Prime number enthusiasts discover new largest prime with 22,338,618 digits.

In this excellent talk the mathematician Vicky Neale gives a fascinating and easy-to-follow introduction to the prime numbers.

Fermat's little theorem and fake primes.

There's been progress on one of the biggest open problems in maths: the twin prime conjecture.

Look for hidden gems in the gaps between known prime numbers.

This year has seen a flurry of results as mathematicians hunt down the elusive proof of the twin prime conjecture. Will they get their wish for Christmas this year?

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.