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:
- $\map f {2, n} = 2^n$
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| \(\ds \map f {3, 4}\) | \(=\) | \(\ds \map f {2, \map f {3, 3} }\) | ||||||||||||
| \(\ds \map f {3, 3}\) | \(=\) | \(\ds \map f {2, \map f {3, 2} }\) | ||||||||||||
| \(\ds \map f {3, 2}\) | \(=\) | \(\ds \map f {2, \map f {3, 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map f {2, 3}\) | $\map f {a, b} = a$ 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): $2^{65,536}$