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2026-04-30T21:54:21Z
User contributions
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https://wiki.bbchallenge.org/w/index.php?title=Champions&diff=880
Champions
2024-09-01T20:46:46Z
<p>Elexunix: /* 2-Symbol TMs */</p>
<hr />
<div>Busy Beaver '''Champions''' are the current record holding [[Turing machine|Turing machines]] who 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 [https://bbchallenge.org/~pascal.michel/bbc Busy Beaver champions] and the [https://bbchallenge.org/~pascal.michel/ha History of Previous Champions].<br />
<br />
== 2-Symbol TMs ==<br />
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 lowerbounds represent the [https://googology.fandom.com/wiki/Fast-growing_hierarchy Fast Growing Hierarchy].<br />
{| class="wikitable"<br />
|+<br />
!<br />
!Runtime<br />
!Champions<br />
!Comment<br />
|-<br />
|[[BB(2)]]<br />
|6<br />
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}<br />
|Discovered and proven by hand by Tibor Radó<br />
|-<br />
|[[BB(3)]]<br />
|21<br />
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}<br />
|Proven by Shen Lin<br />
|-<br />
|[[BB(4)]]<br />
|107<br />
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}<br />
|Discovered and proven by Allen Brady<br />
|-<br />
|[[BB(5)]]<br />
|47,176,870<br />
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}<br />
|Discovered by Heiner Marxen & Jürgen Buntrock in 1989<br />
Proven by [[bbchallenge.org]] in 2024<br />
|-<br />
|[[BB(6)]]<br />
|<math>> 10 \uparrow\uparrow 15</math><br />
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}<br />
|Discovered by Pavel Kropitz in 2022<br />
|-<br />
|[[BB(7)]]<br />
|<br />
|<br />
|<br />
|-<br />
|BB(8)<br />
|<math> > 2 \uparrow^5 4 > f_6(2) </math><br />
|{{TM|1RH1RF_0LC0LH_0RD1LC_0RE1RA_1RB1RE_1RZ1RG_1RF0RE_1LB1LH|halt}}<br />
|Discovered by Racheline in 2024<br />
|-<br />
|BB(9)<br />
|<br />
|<br />
|<br />
|-<br />
|BB(10)<br />
|<math> > f_\omega^2(25) </math><br />
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}<br />
|Discovered by Racheline in 2024<br />
|-<br />
|BB(11)<br />
|<math> > f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9)) </math><br />
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}<br />
|Discovered by Racheline in 2024<br />
|-<br />
|BB(12)<br />
|<math> > f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2)) </math><br />
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}<br />
|Discovered by Racheline in 2024<br />
|-<br />
|BB(13)<br />
|<br />
|<br />
|<br />
|-<br />
|BB(14)<br />
|<math> > f_{\omega + 1}(65536) > g_{64} </math><br />
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}<br />
|Discovered by Racheline in 2024<br />
|-<br />
|BB(15)<br />
|<br />
|<br />
|<br />
|-<br />
|BB(16)<br />
|<math> > f_{\omega + 1}(2 \uparrow\uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow\uparrow 9) </math><br />
|<br />
|Designed by Daniel Nagaj in 2021<ref>Shawn Ligocki. 2022. "BB(16) > Graham's Number". https://www.sligocki.com/2022/07/11/bb-16-graham.html</ref><br />
|-<br />
|BB(51)<br />
|<math> > f_{\epsilon_0 + 1}(8) </math><br />
|<br />
|Discovered by Racheline in 2024<br />
|}<br />
<br />
== 3-Symbol TMs ==<br />
<br />
{| class="wikitable"<br />
|+<br />
!<br />
!Runtime<br />
!Champions<br />
!Comment<br />
|-<br />
|[[BB(2,3)]]<br />
|38<br />
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}<br />
|<br />
|-<br />
|[[BB(3,3)]]<br />
|<math> > 10^{17}</math><br />
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}<br />
|-<br />
|[[BB(4,3)]]<br />
|<math> > 10^{14072} </math><br />
|{{TM|1RB1RZ2RC_2LC2RD0LC_1RA2RB0LB_1LB0LD2RC|halt}}<br />
|<br />
|}<br />
<br />
== 4-Symbol TMs ==<br />
{| class="wikitable"<br />
|+<br />
!<br />
!Runtime<br />
!Champions<br />
!Comment<br />
|-<br />
|BB(2,4)<br />
|<math>\geq3,932,964</math><br />
|<br />
|<br />
|-<br />
|BB(3,4)<br />
|<math>>2\uparrow^{15}5</math><br />
|<br />
|<br />
|}<br />
<br />
== 5-Symbol TMs ==<br />
{| class="wikitable"<br />
|+<br />
!<br />
!Runtime<br />
!Champions<br />
!Comment<br />
|-<br />
|BB(2,5)<br />
|<math>>10^{10^{10^{3314360}}}</math><br />
|<br />
|<br />
|}<br />
<br />
== References ==<br />
<references /></div>
Elexunix