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.

The only known values for \(\Sigma(n)\) are \(\Sigma(1) = 1, \Sigma(2) = 4, \Sigma(3) = 6,\) and \(\Sigma(4) = 13\). The next two numbers are very large, and the known lower bounds are \(\Sigma(5) \geq 4098\) and \(\Sigma(6) \geq 3.514 \cdot 10^{18276}\). Milton Green has proved that \(\Sigma(2n) > 3 \uparrow^k-2 3\), so:

\(\Sigma(10) > 3 \uparrow\uparrow\uparrow 3 \) = tritri

\(\Sigma(12) > 3 \uparrow\uparrow\uparrow\uparrow 3 \) = g1

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.