It is known that $bb(23)$>Graham's number (I do not remember exactly, but $bb(21)$ could already be larger).

But what is the smallest number $n$, such that $bb(n)>f_{\epsilon_0}(5)$ is known ?

Here :

Milton Green's lower bounds of the busy beaver function

it is mentioned that $\Sigma(41,3)$ is much greater than $f_{\epsilon_0}(5)$, but I am looking for a binary machine.

  • 1
    $\begingroup$ Best known for Graham's number is $\Sigma(22)$. I also confirm the result that Deedlit mentions. $\endgroup$ – wythagoras May 29 '16 at 19:14

Wythagoras created an 85-state Turing machine that produced more than $f_{\varepsilon_0}(1907)$ ones, listed here. I believe it is based on an implementation of the Kirby-Paris hydra.

  • $\begingroup$ Very interesting! +1 $\endgroup$ – Peter May 16 '16 at 17:30
  • $\begingroup$ According to the link, $bb(85)$ is even largen than $f_{\epsilon_0}(1907)$ $\endgroup$ – Peter May 16 '16 at 17:38
  • $\begingroup$ whoops, 1905 was a typo. $\endgroup$ – Deedlit May 16 '16 at 18:29

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