Definition:Ackermann-Péter Function

Definition
The Ackermann-Péter function $A: \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{> 0}$ is an integer-valued function defined on the set of ordered pairs of positive integers as:
 * $A \left({x, y}\right) = \begin{cases} y + 1 & : x = 0 \\

A \left({x - 1, 1}\right) & : x > 0, y = 0 \\ A \left({x - 1, A \left({x, y - 1}\right)}\right) & : \text{otherwise} \end{cases}$

Also defined as
Some sources define the Ackermann function as $A: \Z_{> 0} \times \Z_{> 0} \to \Z_{> 0}$ where:


 * $A \left({x, y}\right) = \begin{cases} 2 y & : x = 1 \\

x & : x > 1, y = 1 \\ A \left({x - 1, A \left({x, y - 1}\right)}\right) & : \text{otherwise} \end{cases}$

Examples
No, you're right, I really don't know what I'm doing, at all.

Also known as
The Ackermann-Péter function is also known as the Ackermann function.

However, there are a number of different similar functions which go by this name, so the full appellation can be argued as being more useful.