Busy beaver function

Rado's sigma function (or busy beaver function) \(\Sigma(n)\) is equal to the maximum number of 1s that can be written (in the finished tape) with an n-state, 2-color Turing machine before halting. Turing machines that produce these numbers are called busy beavers.

Rado showed that it the function grows faster than any computable function, and thus it is uncomputable &mdash; determining whether a given Turing machine is a busy beaver takes an infinite number of steps. Despite this, it is possible to evaluate \(\Sigma(n)\) for some small values of \(n\); it has been proven that \(\Sigma(1) = 1\), \(\Sigma(2) = 4\), \(\Sigma(3) = 6,\) and \(\Sigma(4) = 13\). No other values are exactly known, but some lower bounds are \(\Sigma(5) \geq 4098\) and \(\Sigma(6) \geq 3.514 \cdot 10^{18276}\).

Milton Green proved that \(\Sigma(2n) \gg 3 \uparrow^{k-2} 3\), yielding the following lower bounds:

\[\Sigma(8) > 3 \uparrow\uparrow 3 = 7625597484987\]

\[\Sigma(10) > 3 \uparrow\uparrow\uparrow 3 \]

\[\Sigma(12) > 3 \uparrow\uparrow\uparrow\uparrow 3 \]

\(\Sigma(6)\) is already known to be much greater than 7625597484987, so these lower bounds are very weak.

The function's growth rate is comparable to \(f_{\omega^\text{CK}_1}(n)\) in the fast-growing hierarchy, where \(\omega^\text{CK}_1\) is the, the set of all recursive ordinals.