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 $2^{2^8}$.
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This needs considerable tedious hard slog to complete it.
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Sources

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