Footballs, beer bubbles, Fibonacci primes. The most mind-boggling problems yet to be solved by the world's purest mathematicians - and how you can earn $1m if you beat them to it If you know the ...

Prime numbers are the mavericks of maths. They are not evenly divisible by any other number (except 1, which is boring – see last week). They start 2, 3, 5, 7, 11 ...

4 Find the two primes between 15 and 30 for which, if you double them and add 1, the result is still prime. These are called “Sophie Germain primes”. 5 Show why all prime numbers (5 or greater) are ...