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The Busy Beaver function BB (called S originally) was introduced by Tibor Radó in 1962 for 2-symbol Turing machines and later generalized to m-symbol Turing machines:

BB(n, m) = Maximum number of steps taken by a halting n-state, m-symbol Turing machine starting from a blank (all 0) tape

The 2-symbol case BB(n, 2) is abbreviated as BB(n). The Busy Beaver function is not computable, but a few of its values are known:

Small busy beaver values

	2-state	3-state	4-state	5-state	6-state	7-state
2-symbol	BB(2) = 6	BB(3) = 21	BB(4) = 107	BB(5) = 47,176,870	BB(6) > $10\uparrow \uparrow 10\uparrow \uparrow 10\uparrow \uparrow 8$	BB(7) > $3{\uparrow }^{20}3$
3-symbol	BB(2,3) = 38	BB(3,3) $\ge$ 119,112,334,170,342,541	BB(4,3) > $10{\uparrow }^{4}4$		BB(6,3) > $f_{\omega }^2(10\uparrow \uparrow \uparrow 6)$
4-symbol	BB(2,4) = 3,932,964	BB(3,4) > $2{\uparrow }^{15}5$	BB(4,4) > $2{\uparrow }^{509}3$	BB(5,4) > $f_{\omega }^2(10{\uparrow }^{4}10\uparrow \uparrow 65533)$	BB(6,4) > $f_{\omega +1}(10\uparrow \uparrow 65533)$
5-symbol	BB(2,5) > $10^{10^{10^{3314360}}}$	BB(3,5) > $f_{\omega }(2{\uparrow }^{15}5)$	BB(4,5) > $f_{\omega +1}(27041)$	BB(5,5) > $f_{\omega +1}(f_{\omega }^4(6))$	BB(6,5) > $f_{\omega +3}(11)$	BB(7,5) > $f_{\omega +3}^2(10)$
6-symbol	BB(2,6) > $10\uparrow \uparrow 10\uparrow \uparrow 10^{10^{115}}$	BB(3,6) > $f_{\omega }^3(15)$	BB(4,6) > $f_{\omega +2}(29)$	BB(5,6) > $f_{\omega +3}(31)$	BB(6,6) > $f_{\omega \times 2}(36)$	BB(7,6) > $f_{{\omega }^2}(28)$

Table key:

• Orange cell: smallest known Cryptid (a mathematically-hard machine) is in this class
• Tan cell: a Cryptid is known using fewer states or symbols than this class.
• Yellow cell: the Busy Beaver number is known to be more than Graham's number.
• Blank cell: no known champion that surpasses a champion with fewer states or symbols by a non-trivial amount.

1-state domains and 1-symbol domains are omitted in the table as BB(1,m) = 1 and BB(n,1) = n.

About bbchallenge

bbchallenge is a massively collaborative research project whose general goal is to obtain more knowledge on the Busy Beaver function. In practice, it mainly consists in collaboratively building Deciders, programs that automatically prove that some Turing machines do not halt. Other efforts also include:

• Formalizing results using theorem provers (such as Rocq)
• Maintaining Holdouts lists for small busy beaver values
• Proving the behavior of Individual machines
• Finding Cryptids (mathematically-hard machines)
• Searching for new Champions
• Building Accelerated simulators to simulate halting machines faster
• Writing papers and giving talks about the Busy Beaver function, see Papers & Talks
• Exploring different Busy Beaver-style functions built from different systems of computation
• Verify this formula

In June 2024, bbchallenge achieved a significant milestone by proving in Rocq (previously known as Coq) that the 5th busy beaver value, BB(5), is equal to the lower bound found in 1989: 47,176,870.

This Month in Beaver Research (TMBR)

This Month in Beaver Research (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

• TMBR: April 2026
• TMBR: March 2026
• TMBR: February 2026

TMBR: May 2026, TMBR: June 2026, TMBR: July 2026, TMBR: August 2026, TMBR: September 2026, TMBR: October 2026, and TMBR: November 2026 are currently works in progress.

This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, TYBR: 2025 is currently a work in progress.