Definition:Ackermann Function/Mistake 2

Source Work

 * The Dictionary
 * $2^{65536}$
 * $2^{65536}$


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

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:

By induction:
 * $\map f {2, n} = 2^n$

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