Determination of the fifth Busy Beaver value

The bbchallenge Collaboration; Justin Blanchard; Daniel Briggs; Konrad Deka; Nathan Fenner; Yannick Forster; Georgi Georgiev; Matthew L. House; Rachel Hunter; Iijil; Maja Kądziołka; Pavel Kropitz; Shawn Ligocki; mxdys; Mateusz Naściszewski; savask; Tristan Stérin; Chris Xu; Jason Yuen; Théo Zimmermann

arXiv · 2509.12337

Determination of the fifth Busy Beaver value

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Abstract

The Busy Beaver value $S(n)$ is the maximum number of steps that an $n$-state 2-symbol Turing machine can perform from the all-zero tape before halting. $S$ was historically introduced by Tibor Rad\'o in 1962 as one of the simplest examples of an uncomputable function. We prove that $S(5) = 47,176,870$ using the Coq proof assistant. The proof enumerates $181,385,789$ Turing machines with 5 states and, for each machine, decides whether it halts or not. Our result marks the first determination of a new Busy Beaver value in over 40 years and the first Busy Beaver value ever to be formally verified, attesting to the effectiveness of massively collaborative online research (bbchallenge$.$org).

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The bbchallenge Collaboration, Justin Blanchard, Daniel Briggs, Konrad Deka, Nathan Fenner, Yannick Forster, Georgi Georgiev, Matthew L. House, Rachel Hunter, Iijil, Maja Kądziołka, Pavel Kropitz, Shawn Ligocki, mxdys, Mateusz Naściszewski, savask, Tristan Stérin, Chris Xu, Jason Yuen, Théo Zimmermann. 2025-09-15. Determination of the fifth Busy Beaver value. https://arxiv.org/abs/2509.12337

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