Champions: Difference between revisions

Revision as of 20:31, 14 August 2025

Busy Beaver Champions are the current record holding Turing machines which maximize a Busy Beaver function. In this article we focus specifically on the longest running TMs. Some have been proven to be the longest running of all (and so are the ultimate champion) while others are only current champions and may be usurped in the future. For smaller domains, Pascal Michel's website is the canonical source for Busy Beaver champions and the History of Previous Champions.

2-Symbol TMs

Rows are blank if no champion has been found which surpasses a smaller size problem. Also take note that the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{x}(n) } used in the lower bounds represent the Fast-Growing Hierarchy. Note that most champions above 6 states are self-reported and have not been independently verified.


	Runtime	Champions	Discovered By	Verification
BB(2)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6}	`1RB1LB_1LA1RZ` (bbch) `1RB0LB_1LA1RZ` (bbch) `1RB1RZ_1LB1LA` (bbch) `1RB1RZ_0LB1LA` (bbch) `0RB1RZ_1LA1RB` (bbch)	Tibor Radó	Direct Simulation
BB(3)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 21}	`1RB1RZ_1LB0RC_1LC1LA` (bbch)	Proven by Shen Lin	Direct Simulation
BB(4)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 107}	`1RB1LB_1LA0LC_1RZ1LD_1RD0RA` (bbch)	Allen Brady	Direct Simulation
BB(5)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 47\,176\,870}	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch)	Heiner Marxen & Jürgen Buntrock in 1989	Direct Simulation
BB(6)	$>2\uparrow \uparrow \uparrow 5$	`1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE` (bbch)	mxdys in 2025	See mxdys's analysis on the TM page
BB(7)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle > 2 \uparrow^{11} 2 \uparrow^{11} 3}	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)	Pavel Kropitz in 2025	Analyzed by Shawn Ligocki (see TM page)
BB(8)
BB(9)	$>f_{\omega }(f_{9}(2))$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH` (bbch)	Jacobzheng in 2024
BB(10)	$>f_{\omega }^{2}(25)$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ` (bbch)	Racheline in 2024
BB(11)	$>f_{\omega }^{2}(2\uparrow \uparrow 12)>f_{\omega }^{2}(f_{3}(9))$	`1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE` (bbch)	Racheline in 2024
BB(12)	$>f_{\omega }^{4}(2\uparrow \uparrow \uparrow 4-3)>f_{\omega }^{4}(f_{4}(2))$	`0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL` (bbch)	Racheline in 2024
BB(13)
BB(14)	$>f_{\omega +1}(65\,536)>g_{64}$	`1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---` (bbch)	Racheline in 2024
BB(15)	$>f_{\omega +1}(f_{\omega }(10^{57}))$	`0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN` (bbch)	Jacobzheng in 2025
BB(16)	$>f_{\omega +1}^{2}(10^{10^{57}})$	User:Jacobzheng/BB(16)	Jacobzheng in 2025
BB(17)
BB(18)	$>f_{\omega +2}(f_{\omega +1}^{3}(f_{\omega }^{2}(60)))$	User:Jacobzheng/BB(18)	Jacobzheng in 2025
BB(19)
BB(20)	$>f_{\omega +2}^{2}(21)$		Racheline in 2024
BB(21)	$>f_{\omega ^{2}}^{2}(4\uparrow \uparrow 341)$		Racheline in 2024
BB(40)	$>f_{\omega ^{\omega }}(75\,500)$	User:Jacobzheng/BB(40)	Jacobzheng in 2024
BB(41)	$>f_{\omega ^{\omega }}^{4}(32)$	User:Jacobzheng/BB(41)	Jacobzheng in 2024
BB(51)	$>f_{\varepsilon _{0}+1}(8)$		Racheline in 2024

3-Symbol TMs


	Runtime	Champions	Discovered By	Verification
BB(2,3)	$38$	`1RB2LB1RZ_2LA2RB1LB` (bbch)	Allen Brady in 1988	Direct Simulation
BB(3,3)	$>10^{17}$	`0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC` (bbch)	Terry & Shawn Ligocki in 2007	Analysis by Pascal Michel
BB(4,3)	$>2\uparrow \uparrow \uparrow 2^{2^{32}}$	`0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD` (bbch)	Pavel Kropitz in 2024

4-Symbol TMs


	Runtime	Champions	Discovered By	Verification
BB(2,4)	$3\,932\,964$	`1RB2LA1RA1RA_1LB1LA3RB1RZ` (bbch)	Terry & Shawn Ligocki in 2005	Pascal Michel, Heiner Marxen, Allen Brady
BB(3,4)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle > 2 \uparrow^{15} 5}	`1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC` (bbch)	Pavel Kropitz in 2024	Analysis by Shawn Ligocki

5-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,5)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle > 10^{10^{10^{3\,314\,360}}}}	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ` (bbch)	Daniel Yuan in 2024	mxdys in Coq
BB(3,5)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle > f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)}	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC` (bbch)	Racheline in 2024

6-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,6)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle > 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}}	`1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA` (bbch)	Pavel Kropitz in 2023	Analysis by Shawn Ligocki

@@ Line 2: / Line 2: @@
 == 2-Symbol TMs ==
-Rows are blank if no champion has been found which surpasses a smaller size problem. Take also note that the <math> f_{x}(n) </math> used in the lower bounds represent the [[Fast-Growing Hierarchy]]. Note that most champions above 6 states are self-reported and have not been independently verified.
+Rows are blank if no champion has been found which surpasses a smaller size problem. Also take note that the <math> f_{x}(n) </math> used in the lower bounds represent the [[Fast-Growing Hierarchy]]. Note that most champions above 6 states are self-reported and have not been independently verified.
 {| class="wikitable"