The Busy Beaver Challenge, a notoriously difficult question in theoretical computer science, is now producing answers so large they're impossible to write out using standard mathematical notation. Imagine that someone gives you a list of five numbers: 1, 6, 21, 107 and--wait for it--47,176,870. Can you guess what comes next? These are the first five busy beaver numbers. They form a sequence that's intimately tied to one of the most notoriously difficult questions in theoretical computer science. Determining the values of busy beaver numbers is a daunting challenge that has attracted a cult following among both professional and amateur mathematicians for over 60 years. Researchers identified the first four busy beaver numbers in the 1960s and 1970s.

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.