Editing Busy beaver (section)

===Exact values and lower and upper bounds===
The following table lists the exact values and some known lower bounds for ''S''(''n''), Σ(''n''), and several other busy beaver functions. In this table, 2-symbol Turing machines are used. Entries listed as "?" are at least as large as other entries to the left (because all n-state machines are also (n+1) state machines), and no larger than entries above them (because S(n) ≥ space(n) ≥ Σ(n) ≥ num(n)). So, space(6) is known to be greater than 10⇈15, as space(n) ≥ Σ(n) and Σ(6) > 10⇈15. {{val|47176870}} is an upper bound for space(5), because S(5) = {{val|47176870}} (<ref name=":4" />) and S(n) ≥ space(n). 4098 is an upper bound for num(5), because Σ(5) = 4098 (<ref name=":8" />) and Σ(n) ≥ num(n). The last entry listed as "?" is num(6), because Σ(6) > 10⇈15, but Σ(n) ≥ num(n).
{| class="wikitable" style="text-align:right"
|+ Values of busy beaver functions
! Function
! colspan="2" | 2-state
! colspan="2" | 3-state
! colspan="2" | 4-state
! colspan="2" | 5-state
! colspan="2" | 6-state
|-
! S(n)
|   6              || <ref name=":62" />
|  21              || <ref name=":62" />
| 107              || <ref name=":62" />
| {{val|47176870}} || <ref name=":4" />
| > 10⇈15         || <ref name=":5" />
|-
! space(n)
| 4 || <ref name=":8" />
| 7 || <ref name=":8" />
| 16 || <ref name=":8" />
| ≥ {{val|12289}} <br>≤ {{val|47176870}}<br>S(n) ≥ space(n) || <ref name=":8" />
| > 10⇈15<br>space(n) ≥ Σ(n) ||
|-
! Σ(n)
| 4 || <ref name=":8" />
| 6 || <ref name=":8" />
| 13 || <ref name=":8" />
| 4098 || <ref name=":8" />
|> 10⇈15 || <ref name="ligocki_bb6" />
|-
! num(n)
|4 ||<ref name=":8" />
|6 ||<ref name=":8" />
|12 ||<ref name=":8" />
|≤ 4098<br>Σ(n) ≥ num(n) ||
|? ||
|}

The 5-state busy beaver was discovered by Heiner Marxen and Jürgen Buntrock in 1989, but only proved to be the winning fifth busy beaver — stylized as BB(5) — in 2024 using a proof in [[Coq (software)|Coq]].<ref>{{Cite web |date=2024-07-02 |title=[July 2nd 2024] We have proved "BB(5) = 47,176,870" |url=https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237 |access-date=2024-07-02 |website=The Busy Beaver Challenge |language=en |archive-date=2024-07-02 |archive-url=https://web.archive.org/web/20240702145239/https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237 |url-status=live }}</ref><ref>{{Cite web |title=The Busy Beaver Challenge |url=https://bbchallenge.org/ |access-date=2024-07-02 |website=bbchallenge.org |language=en |archive-date=2024-07-02 |archive-url=https://web.archive.org/web/20240702132423/https://bbchallenge.org/ |url-status=live }}</ref>