Editing Busy beaver (section)

==== Notable examples ====
* A 745-state binary Turing machine has been constructed that halts [[if and only if]] [[ZFC]] is inconsistent.<ref name="AaronsonJuly2023" /><ref name=":3" />

* A 744-state Turing machine has been constructed that halts if, and only if, the [[Riemann hypothesis]] is false.<ref name="Aaronson3" /><ref name=":1" />

* A 43-state Turing machine was constructed that halts if, and only if, [[Goldbach's conjecture]] is false. This was further reduced to 25-state machine,<ref name="Aaronson3" /><ref name=":1" /> and later formally proved and verified in the [[Lean 4]] theorem proving language.<ref>{{cite web|last=Leng|first=Yijun|title=GitHub Repository "goldbach_tm27"|website=[[GitHub]] |url=https://github.com/lengyijun/goldbach_tm}}</ref>

* A 15-state Turing machine has been constructed that halts if and only if the following conjecture formulated by [[Paul Erdős]] in 1979 is false: for all ''n'' > 8 there is at least one digit 2 in the base 3 representation of 2<sup>''n''</sup>.<ref>{{cite report |title=On the hardness of knowing busy beaver values BB(15) and BB(5,4) |author=Tristan Stérin and Damien Woods |date=July 2021 |arxiv=2107.12475 |institution=Maynooth University |type=Technical Report}}</ref><ref>{{cite journal |last=Erdös |first=Paul |author-link=Paul Erdős |date=1979 |title=Some unconventional problems in number theory. |url=https://jstor.org/stable/2689842 |journal=[[Math. Mag.]] |volume=52 |issue=2 |pages=67–70 |doi=10.1080/0025570X.1979.11976756 |jstor=2689842 |archive-date=2022-06-13 |access-date=2022-07-07 |archive-url=https://web.archive.org/web/20220613231912/https://www.jstor.org/stable/2689842 |url-status=live }}</ref>