# Definition:Ackermann Function/Mistake 2

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## Source Work

1986: David Wells: *Curious and Interesting Numbers*:

- The Dictionary
- $2^{65536}$

1997: David Wells: *Curious and Interesting Numbers* (2nd ed.):

- The Dictionary
- $2^{65,536}$

## Mistake

*Ackermann's function is defined by $\map f {a, b} = \map f {a - 1, \map f {a, b - 1} }$ where $\map f {1, b} = 2 b$ and $\map f {a, 1} = a$ for $a$ greater than $1$.**$\map f {3, 4} = 2^{65,536}$, which has more than $19,000$ digits.*

In fact, what we find is as follows.

Let us define $f$ as above:

- $\map f {a, b} = \begin{cases} 2 b & : a = 1 \\ a & : a > 1, b = 1 \\ \map f {a - 1, \map f {a, b - 1} } & : \text{otherwise} \end{cases}$

Then we have:

\(\ds \map f {2, 3}\) | \(=\) | \(\ds \map f {1, \map f {2, 2} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2 \map f {2, 2}\) | $\map f {a, b} = 2 b$ when $a = 1$ | |||||||||||

\(\ds \) | \(=\) | \(\ds 2 \map f {1, \map f {2, 1} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 2 \map f {2, 1}\) | $\map f {a, b} = 2 b$ when $a = 1$ | |||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 2 \times 2\) | $\map f {a, b} = a$ when $b = 1$ | |||||||||||

\(\ds \) | \(=\) | \(\ds 8\) |

By induction:

- $\map f {2, n} = 2^n$

This needs considerable tedious hard slog to complete it.In particular: The above statement needs to be demonstrated.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. |

\(\ds \map f {3, 4}\) | \(=\) | \(\ds \map f {2, \map f {3, 3} }\) | ||||||||||||

\(\ds \map f {3, 3}\) | \(=\) | \(\ds \map f {2, \map f {3, 2} }\) | ||||||||||||

\(\ds \map f {3, 2}\) | \(=\) | \(\ds \map f {2, \map f {3, 1} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map f {2, 3}\) | $\map f {a, b} = a$ when $b = 1$ | |||||||||||

\(\ds \) | \(=\) | \(\ds 8\) | from above | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map f {3, 3}\) | \(=\) | \(\ds \map f {2, 8}\) | from above | ||||||||||

\(\ds \) | \(=\) | \(\ds 2^8\) | from above | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map f {3, 4}\) | \(=\) | \(\ds \map f {2, 2^8}\) | from above | ||||||||||

\(\ds \) | \(=\) | \(\ds 2^{\map f {2^8} }\) | from above | |||||||||||

\(\ds \) | \(=\) | \(\ds 2^{256}\) |

and not $2^{65 \, 536}$ after all.

## Sources

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