Mathematicians spent 2025 exploring the edge of mathematics

In 2025, the edges of mathematics came a little more sharply into view when members of the online Busy Beaver Challenge community closed in on a huge number that threatens to defy the logical underpinnings of the subject.

This number is the next in the “Busy Beaver” sequence, a series of ever-larger numbers that emerges from a seemingly simple question – how do we know if a computer program will run forever?

To find out, researchers turn to the work of mathematician Alan Turing, who showed that any computer algorithm can be mimicked by imagining a simplified device called a Turing machine. More complex algorithms correspond to Turing machines with larger sets of instructions or, in mathematical parlance, more states.

Each Busy Beaver number BB(n) captures the longest possible run-time for a Turing machine with n states. For example BB(1) is 1 and BB(2) is 6, so making the algorithm twice as complex increases its runtime sixfold. But the rate of this increase turns out to be extreme, for example, the fifth Busy Beaver number is 47,176,870.

Members of the Busy Beaver Challenge pinned down the exact value of BB(5) in 2024, which ended a 40-year effort to study all Turing machines with five states. So, naturally, 2025 was marked by a collective chase after BB(6).

In July, a member known as mxdys discovered a lower limit on its size, and that number turned out not only to be much bigger than BB(5) but truly enormous even when compared with the number of particles in our universe.

Writing down all of its digits is physically impossible, so mathematicians use a kind of notation called tetration instead. This is equivalent to repeatedly raising a number to a higher power, for example, 2 tetrated to 2 is equal to 2 raised to the power of 2 raised to the power of 2, which is 16. BB(6) is at least 2 tetrated to 2 tetrated to 2 tetrated to 9, a gargantuan tower of iterated tetration.

Pinning down BB(6) won’t just be a matter of setting records, but it may also have deep implications for all of mathematics. This is because Turing proved that there must be some Turing machines whose behaviour cannot be predicted under a set of axioms called ZFC theory, which forms the foundation on which all standard modern mathematics stands.

Already, researchers have proven that BB(643) would elude ZFC theory, but whether this could happen for smaller numbers is an open question – one that the Busy Beaver Challenge may contribute to answering.

In July, there were 2728 Turing machines that have six states but whose stopping behaviour had not yet been checked. By October that number dropped to 1618. “The community is being super active at the moment,” says computer scientist Tristan Stérin, who launched the Busy Beaver Challenge in 2022.

One of the holdout machines could hold the key to the exact value of BB(6). One of them could also turn out to be unknowable, exposing the bounds of the ZFC framework and much of modern mathematics. Over the course of the next year, mathematics enthusiasts across the globe will certainly be hard at work trying to understand them all.