Mathematicians spent 2025 exploring the edge of mathematics

In 2025, the boundaries of mathematics became more apparent when members of the online community Busy Beaver Challenge Community closed on huge amount it threatens to challenge the logical basis of the subject.

This number is the next in the Busy Beaver sequence, a series of increasingly larger numbers arising from a seemingly simple question: How do we know if a computer program will run forever?

To find out, researchers turn to the works of mathematician Alan Turingwho showed that any computer algorithm can be imitated by introducing a simplified device called a Turing machine. More complex algorithms correspond to Turing machines with a larger set of instructions or, in mathematical terms, a larger number of states.

Each Busy Beaver BB(n) number represents the maximum possible running time of a Turing machine with n states. For example, BB(1) is 1 and BB(2) is 6, so doubling the complexity of an algorithm increases its execution time by a factor of six. But the speed of this increase is extreme, for example, the fifth number of Busy Beaver is 47,176,870.

Busy Beaver Challenge participants detained exact value BB(5) in 2024, ending a 40-year effort to understand all five-state Turing machines. So naturally, 2025 marked the collective pursuit of BB(6).

In July, the member known as mxdys has discovered the lower limit of its sizeand this number turned out to be not only much larger than BB(5), but also truly huge even compared to the number of particles in our Universe.

It is physically impossible to write down all of its digits, so mathematicians instead use a form of notation called tetration. This is equivalent to repeatedly raising a number to a higher power, for example, 2 tetraded to 2 is 2 raised to the power of 2 raised to the power of 2, which is 16. BB(6) is at least 2 tetraded to 2, to 2 tetrated to 9, a giant tower of re-tetradization.

Establishing BB(6) will not just be a matter of setting records, but could also have profound implications for all of mathematics. This is because Turing proved that there must be Turing machines whose behavior cannot be predicted by a set of axioms called ZFC theory.which forms the foundation on which all standard modern mathematics stands.

Researchers have already proven that BB(643) will escape the ZFC theory, but whether this can happen for smaller numbers is an open question that the Busy Beaver Challenge can help answer.

In July, there were 2,728 six-state Turing machines whose stopping behavior had not yet been tested. By October, that number had dropped to 1,618. “The community is very active right now,” the computer scientist says. Tristan Sterinwhich launched the Busy Beaver Challenge in 2022.

One of the holding machines may hold the key to the exact value of BB(6). One of them may also be unknowable, exposing the boundaries of the ZFC structure and much of modern mathematics. Over the next year, math enthusiasts around the world are sure to be hard at work trying to understand them all.