Definition:Ackermann Function/Mistake 1

Source Work

 * The Dictionary
 * $65,536$
 * $65,536$

Mistake

 * The Ackermann function is one of the fastest increasing functions used in mathematics. Its values from $f \left({0}\right)$ to $f \left({5}\right)$ are $1$, $3$, $4$, $8$, $65,536$.

First note that this should read:
 * Its values from $f \left({0}\right)$ to $f \left({4}\right)$ are $1$, $3$, $4$, $8$, $65,536$.

Next it should be noted that has at this stage not specified what is meant by the Ackermann function. When later in he does define it, he does so as a function of $2$ variables.

It is probable that he was referring to the function:
 * $f \left({n}\right) = H_n \left({2, n}\right)$

where $H_n$ is the $n$th hyperoperator, defined as:

$H_n \left({x, y}\right) = \begin{cases} y + 1 & : n = 0 \\ x & : n = 1, y = 0 \\ 0 & : n = 2, y = 0 \\ 1 & : n > 2, y = 0 \\ H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$

This leads to the results:

To simplify what happens later, let us check $H_2 \left({2, 1}\right)$.

We have:

Then: