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 with an n-state, 2-color Turing machine before halting. Turing machines that produce these numbers are called busy beavers.

Rado's sigma function is uncomputable &mdash; determining whether a given Turing machine is a busy beaver is an undecidable problem. It grows extremely fast; Rado showed that it grows faster than any computable function. However, for some small values of \(n\), \(\Sigma(n)\) can be evaluated: \(\Sigma(1) = 1, \Sigma(2) = 4, \Sigma(3) = 6,\) 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 \(\Sigma(2n) >> 3 \uparrow^{k-2} 3\), 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.