If by ‘simplest’ you mean easiest to explain, then it’s arguably the so-called ‘Twin Prime Conjecture’. Even schoolchildren can understand it, but proving it has so far defeated the world’s best mathematicians.

Prime numbers are the building blocks from which every whole number can be made. All numbers are thus either prime themselves, or can be made from a unique combination of primes multiplied together. When the prime numbers are written down (2, 3, 5, 7, 11, 13, 17, 19, and so on) two patterns emerge. First, they become progressively rare: while 25 per cent of numbers between 1 and 100 are prime, this falls to just 5 per cent between 1 and a billion. But while they thin out, there still seems to be an endless supply of ‘twin primes’ like 3 and 5, 29 and 31, 41 and 43, which differ by 2. But do these twins ever run out?

Over 2,300 years ago, the Greek mathematician Euclid proved that primes themselves go on forever. So it seems possible that twin primes might do so too. That’s not proof, however – and this remains elusive. Currently, all that mathematicians have managed to prove is that there’s an infinite supply of primes differing by no more than 246.

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## Authors

Robert is a science writer and visiting professor of science at Aston University.