The Millennium Prize Problems are of particular interest, as there is a $1 million prize for solving each of the seven problems posed. So far, only one, the Poincaré Conjecture, has been solved.

Another interesting group of problems is Hilbert’s 24 Problems. These are problems that David Hilbert put forward as the most important unsolved problems in mathematics. There’s no prize for solving any of them.

Perhaps the most famous problem in mathematics is Fermat’s Last Theorem (FLT). Fermat wrote in the margins of a book that a^n+b^n=c^n had no solutions for n>2, but that there wasn’t enough space for the solution. The formula is the general form of Pythagoras’ Theorem ( a^2+b^2=c^2), which elementary students can understand: and yet this proof eluded some of the finest mathematical minds for over 300 years! It was finally solved by Andrew Wiles, who spent 30 years working in secret on the proof (Simon Singh has written a readable and rather thrilling* book on the proof). It is, however, unlikely that Fermat’s ‘trivial’ proof was correct: some of Wiles’ methods were not available in Fermat’s time. Some have suggested that perhaps Fermat did have a simple proof: the science fiction writer, Arthur C. Clarke, wrote a novel about a mathematician who finds one.

My favourite unsolved proof is on both the Hilbert and Millennium Prize lists: Riemann’s Hypothesis. I did my pre-teacher training thesis on this hypothesis, which relates to the distribution of prime numbers. It’s mind-bendingly beautiful, it sings in eldritch tones, it’s a saraband in an alien landscape. It’s too exquisite not to be true, and too terrifying to be real. If it is ever solved, online security is potentially gone – it’s based on prime numbers. This isn’t just a problem for online banking and Paypal – it’s the whole international system used by banks to transfer funds. We might have to go back to bartering. I can recommend the books by both Karl Sabbagh and Marcus du Sautoy for introducing the hypothesis to an informed layperson.

* – For a given value of thrilling. I found it tremendously exciting, but I’m odd like that.