Ackermann-Péter Function/Examples

Examples of Ackermann-Péter Function

The Ackermann-Péter function $A: \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{> 0}$ is defined as:

$\map A {m, n} = \begin{cases} n + 1 & : m = 0 \\

\map A {m - 1, 1} & : m > 0, n = 0 \\ \map A {m - 1, \map A {m, n - 1} } & : \text{otherwise} \end{cases}$

$\begin {array} {c|c|c|c}

\map A {m, n} & m = 0 & m = 1 & m = 2 & m = 3 \\ \hline n = 0 & 1 & \map A {0, 1} & \map A {1, 1} & \map A {2, 1} \\ n = 1 & 2 & \map A {0, \map A {1, 0} } & \map A {1, \map A {2, 0} } & \map A {2, \map A {3, 0} } \\ n = 2 & 3 & \map A {0, \map A {1, 1} } & \map A {1, \map A {2, 1} } & \map A {2, \map A {3, 1} } \\ n = 3 & 4 & \map A {0, \map A {1, 2} } & \map A {1, \map A {2, 2} } & \map A {2, \map A {3, 2} } \\ n = 4 & 5 & \map A {0, \map A {1, 3} } & \map A {1, \map A {2, 3} } & \map A {2, \map A {3, 3} } \\ \end{array}$

which leads to:

$\begin{array}{c|c|c|c}

\map A {m, n} & m = 0 & m = 1 & m = 2 & m = 3 \\ \hline n = 0 & 1 & 2 & 3 & 5 \\ n = 1 & 2 & 3 & 5 & 13 \\ n = 2 & 3 & 4 & 7 & \map A {2, 13} \\ n = 3 & 4 & 5 & 9 & \map A {2, \map A {3, 2} } \\ n = 4 & 5 & 6 & 11 & \map A {2, \map A {3, 3} } \\ \end{array}$

This needs considerable tedious hard slog to complete it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.