I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related to these functions, but I am not sure if they still keep the property of being NOT primitive recursive. My intuition is that they are, but I am not sure. Any textbook or references to read to get a better understanding of this will be very appreciated (and a straight response too!).
NOTE: I read this related question, but it doesnt help me.
If I have understood well yes.
Because if follow from the definiton of hyperoperators definition.
$H_0(a,b):=S(b)$
is the successor function, and every hyperoperation is defined recursively from the successor function:
(recursive definition of $H_{n+1}$ using $H_n$)
$i)$ $H_{n+1}(a,0):=a_{n+1}$
$ii)$ $H_{n+1}(a,b+1):=H_n(a,H_{n+1}(a,b))$
Here $a_{n+1}$ is the initial value of the function when the argument is $0$ and in the case of hyporoperators we have that
$a_ {n+1}:= \begin{cases} a, & \text{if $n=0$} \\ 0, & \text{if $n=1$ } \\ 1, & \text{if $n\gt 1$ } \\ \end{cases}$
Since $H_0(a,b)=S(b)$ is the successor funtion and is basic primitive rucursive funtion, we have that all the hyperoperations (that are defined from $H_0(a,b)=S(b)$) are Primitve recursive functions.