# Ackermann Function/Examples

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## Examples of Ackermann Function

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

- $\map A {m, n} = \begin{cases} 2 n & : m = 1 \\ m & : m > 1, n = 1 \\ \map A {m - 1, \map A {m, n - 1} } & : \text{otherwise} \end{cases}$

- $\begin{array}{c|c|c|c} \map A {m, n} & m = 1 & m = 2 & m = 3 & m = 4 & \cdots & m = k \\ \hline n = 1 & 2 & 2 & 3 & 4 & & k \\ n = 2 & 4 & \map A {1, \map A {2, 1} } & \map A {2, \map A {3, 1} } & \map A {3, \map A {4, 1} } & & \map A {k - 1, \map A {k, 1} } \\ n = 3 & 6 & \map A {1, \map A {2, 2} } & \map A {2, \map A {3, 2} } & \map A {3, \map A {4, 2} } & & \map A {k - 1, \map A {k, 2} } \\ n = 4 & 8 & \map A {1, \map A {2, 3} } & \map A {2, \map A {3, 3} } & \map A {3, \map A {4, 3} } & & \map A {k - 1, \map A {k, 3} } \\ n = 5 & 10 & \map A {1, \map A {2, 4} } & \map A {2, \map A {3, 4} }\ & \map A {3, \map A {4, 4} } & & \map A {k - 1, \map A {k, 4} } \\ \vdots & & & & & & \\ n = j & 2 j & \map A {1, \map A {2, j - 1} } & \map A {2, \map A {3, j - 1} } & \map A {3, \map A {4, j - 1} } & & \map A {k - 1, \map A {k, j - 1} } \\ \end{array}$

which leads to:

- $\begin{array}{c|c|c|c} \map A {m, n} & m = 1 & m = 2 & m = 3 & m = 4 & \cdots & m = k \\ \hline n = 1 & 2 & 2 & 3 & 4 & & k \\ n = 2 & 4 & 4 & 8 & \map A {3, 4} & & \map A {k - 1, k} \\ n = 3 & 6 & 8 & 2^8 & \map A {3, \map A {4, 2} } & & \map A {k - 1, \map A {k, 2} } \\ n = 4 & 8 & 16 & 2^{2^8} & \map A {3, \map A {4, 3} } & & \map A {k - 1, \map A {k, 3} } \\ n = 5 & 10 & 32 & \map A {2, \map A {3, 4} } & \map A {3, \map A {4, 4} } & & \map A {k - 1, \map A {k, 4} } \\ \vdots & & & & \\ n = j & 2 j & 2^j & \map A {2, \map A {3, j - 1} } & \map A {3, \map A {4, j - 1} } & & \map A {k - 1, \map A {k, j - 1} } \\ \end{array}$

There is believed to be a mistake here, possibly a typo.In particular: This needs to be checked, because "in the book" it has that $\map A {3, 4}$ works out at $2^{65,536} = 2^{2^{16} }$ not $2^{2^8}$.You can help ProofWiki by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all.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 `{{Mistake}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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. |

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2^{65536}$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2^{65,536}$