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Let $\mathrm{BB}:\mathbb{N}\to\mathbb{N}$ be the busy beaver function. Eventually $\mathrm{BB}(n)>n^2$ so the sum$$S=\sum_{n=1}^\infty \frac{1}{\mathrm{BB}(n)}$$converges.

Question. What is the largest rational number $q$ such that $S\geq q$ is not known to be inconsistent with ZFC?

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    $\begingroup$ Which definition of BB(n) are you using? $\endgroup$
    – JoshuaZ
    May 4, 2021 at 11:29
  • $\begingroup$ @JoshuaZ the definition in scottaaronson.com/papers/bb.pdf on page 2 $\endgroup$
    – reyl
    May 4, 2021 at 11:31
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    $\begingroup$ Since BB(5) is not known, probably 1/BB(1)+1/BB(2)+1/BB(3)+1/BB(4)+1/(our best bound for BB(5)) $\endgroup$
    – Wojowu
    May 4, 2021 at 11:33
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    $\begingroup$ "Eventually $BB(n) > n^2$ . . . " Yes, that's putting it mildly. :) $\endgroup$
    – Will Brian
    May 4, 2021 at 12:38
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    $\begingroup$ Adding on to Wojowu's comment slightly: thanks to how fast $BB(n)$ grows, we know that the first several decimal places of this constant are $1.2236315$. The next (at least) 36000 decimal places depend only on the value of $BB(5)$, which we do not yet know. $\endgroup$
    – Nathaniel Johnston
    May 4, 2021 at 13:26

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