User:MrSolis/Playground

5-state busy beaver winner (Remake)

The 5-state busy beaver (BB(5)) winner is 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch). Discovered by Heiner Marxen and Jürgen Buntrock in 1989, this machine proved that $\mathrm{BB}(5)\ge 47176870\phantom{}$ and $\mathrm{\Sigma }(5)\ge 4098$ at the time.

Analysis

Let $g(x):=0^{\mathrm{\infty }}\text{<mrow data-mjx-texclass="ORD"><mo><</mo>A</mrow>}{1}^x{0}^{\mathrm{\infty }}$. Then, $$\begin{array}{lll}g(3x)& \phantom{}\stackrel{5x^2+19x+15}{\to }& g(5x+6),\\ g(3x+1)& \phantom{}\stackrel{5x^2+25x+27}{\to }& g(5x+9),\\ g(3x+2)\phantom{}& \stackrel{6x+12}{\to }& 0^{\mathrm{\infty }}1\text{<mrow data-mjx-texclass="ORD">Z<mo>></mo></mrow>}01001^{x+1}10^{\mathrm{\infty }}\end{array}$$