Ackermann function

In computability theory, the Ackermann function or Ackermann-Péter function is a simple example of a computable function that is not primitive recursive. The set of primitive recursive functions is a subset of the set of general recursive functions. Ackermann's function is an example that shows that the former is a strict subset of the latter. .

Definition
The Ackermann function is defined recursively for non-negative integers m and n as follows::


 * $$ A(m, n) = \begin{cases}  n+1 & \mbox{if } m = 0 \\  A(m-1, 1) & \mbox{if } m > 0 \mbox{ and } n = 0 \\  A(m-1, A(m, n-1)) & \mbox{if } m > 0 \mbox{ and } n > 0.  \end{cases} $$

Rapid growth
Its value grows rapidly; even for small inputs, for example A(4,2) contains 19,729 decimal digits.

Holomorphic extensions
The lowest Ackermann functions allow the extension to the complex values of the second argument. In particular,
 * $$ A(0,z) =z+1$$
 * $$ A(1,z) =z+2=2+(n\!+\!3)-3$$
 * $$ A(2,z) =2z+3=2\!\cdot\!(n\!+\!3)-3$$

The 3th Ackermann function is related to the exponential on base 2 through
 * $$ A(3,z) = \exp_2(z\!+\!3)-3=2^{z+3}-3$$

The 4th Ackermann function is related to tetration on base 2 through
 * $$ A(4,z)=\mathrm{tet}_2(z+3)-3$$

which allows its holomorphic extension for the complex values of the second argument.

For $$n>4$$ no holomorphic extension of $$A(n,z)$$ to complex values of $$z$$ is yet reported.