We prove that $S(5) = 47,176,870$ using the Coq proof assistant. 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, and $S$ was historically introduced by Tibor Radó in 1962 as one of the simplest examples of an uncomputable function. 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 determinati...
