Definition:Ackermann Function/Mistake 2

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): $2^{65,536}$