Some numbers are so unimaginably large that they defy the bounds of modern mathematics, and now mathematicians are closing in on a number that may mark the edge of this bizarre abyss
By Karmela Padavic-Callaghan
7 July 2025
What lurks at the edge?
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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.
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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. The longest possible run-time for each number of states is called the Busy Beaver number or BB(n), and this sequence grows incredibly quickly: BB(1) is 1, BB(2) is 6, but the fifth Busy Beaver number is 47,176,870.
The exact value of the next Busy Beaver number, the sixth, is unknown, but an online community called the Busy Beaver Challenge is attempting to discover it – they uncovered BB(5) in 2024, putting an end to a 40-year search. Now, a member known as “mxdys” has discovered it must be at least as big as a number that is so large that even describing it requires some explanation.
“This number is so far beyond physical, it’s not even funny,” says Shawn Ligocki, a software engineer and Busy Beaver Challenge contributor. He compares the search through all the possible Turing machines to fishing in some deep mathematical sea where only odd, exotic bits of code swim in the dark.