In the book that I am reading there is the following proof:
To prove that Ackermann's function is not primitive recursive, we need the following properties concerning the values of $A$.
1. $A(x,y)>y$.
2. $A(x,y+1)>A(x,y)$.
3. If $y_2>y_1$, then $A(x,y_2)>A(x,y_1)$.
4. $A(x+1, y) \geq A(x,y+1)$.
5. $A(x,y)>x$.
6. If $x_2>x_1$, then $A(x_2, y)>A(x_1, y)$.
7. $A(x+2, y)>A(x,2y)$.
We will prove that Ackermann's function is not primitive recursive by showing that it "grows faster" than any primitive recursive function. What we shall attempt to do is show that for any given $n-$variabl…