Definition:Ackermann Function/Mistake 2

Source Work

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

Mistake

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

In fact, what we find is as follows.

Let us define $f$ as above:


 * $f \left({a, b}\right) = \begin{cases} 2 b & : a = 1 \\

a & : a > 1, b = 1 \\ f \left({a - 1, f \left({a, b - 1}\right)}\right) & : \text{otherwise} \end{cases}$

Then we have:

By induction:
 * $f \left({2, n}\right) = 2^n$

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