Mathematicians are chasing a number that may reveal the edge of maths

Amateur mathematicians are closing in on an unimaginably huge number – one so large that it brushes up on the edge of what is even knowable within the framework of modern mathematics. It all stems from a seemingly simple question: how do you know if a computer program will run forever? Answering this starts with mathematician Alan Turing. In the 1930s, he showed that any computer algorithm can be mimicked by imagining a simple "Turing machine" that reads and writes 0s and 1s on an infinitely long tape by following a set of instructions called states, with more complex algorithms requiring more states. For every number of states, such as 5 or 100, there are finitely many corresponding Turing machines, but it is unclear for how long each of these machines must run.