jmc

We iterate through the definition of $A(m,n).$ Each step below is marked either $(1),$ $(2),$ or $(3),$ corresponding to the three parts of the definition of $A(m,n)$: $$\begin{array}{rl}A(2,1)& \stackrel{(3)}{=}A(1,A(2,0))\\ & \stackrel{(2)}{=}A(1,A(1,1))\\ & \stackrel{(3)}{=}A(1,A(0,A(1,0)))\\ & \stackrel{(2)}{=}A(1,A(0,A(0,1)))\\ & \stackrel{(1)}{=}A(1,A(0,2))\\ & \stackrel{(1)}{=}A(1,3)\\ & \stackrel{(3)}{=}A(0,A(1,2))\\ & \stackrel{(3)}{=}A(0,A(0,A(1,1))).\end{array}$$In the last few steps, we actually computed $A(1,1)=3,$ so we have $$A(2,1)=A(0,A(0,3))=A(0,4)=\boxed{5}$$by applying $(1)$ twice.

(Note: the function $A(m,n)$ is called the Ackermann function. Because of its deeply recursive definition, $A(m,n)$ grows extremely quickly. For example, other values of $A(m,n)$ include $A(3,3)=29$ and $A(4,2)=2^{65536}-3,$ which has tens of thousands of digits in base ten!)