TITLE: Are hyperoperators primitive recursive?
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: (Elaborating on my comment to the accepted answer ...) 
For each $n \in \mathbb{N}$, the $n$th hyperoperation 
$$H_n: \mathbb{N}^2 \to \mathbb{N}: (x,y) \mapsto H_n(x,y)$$ 
is primitive recursive.
However, for any integer constant $c\ge 2$, the following functions are not primitive recursive:
$$\begin{align}
f_c&: \mathbb{N} \to \mathbb{N}: n \mapsto H_n(c,n)\\
g_c&: \mathbb{N}^2 \to \mathbb{N}: (n,y) \mapsto H_n(c,y)\\
h&: \mathbb{N}^3 \to \mathbb{N}: (n,x,y) \mapsto H_n(x,y).
\end{align}$$
When $n$ is included as an argument, the function grows too fast to be primitive recursive.
E.g., $x\uparrow^n y\ $ is primitive recursive as a function of $(x,y)$ for fixed $n$, but it is not primitive recursive as a function of $(n,y)$ for fixed $x\ge 2$. 
Essentially these results are proved in Chapter 13 of  "Computability, Complexity, and Languages" (1983) by Davis & Weyuker, and also sketched in Introduction to Computability Theory by Zucker & Pretorius.