Ackermann Function primitive recursive

I am reading the wikipedia page on ackermann's function, http://en.wikipedia.org/wiki/Ackermann_function

And I am having trouble understanding WHY ackermann's function is an example of a function which is not primitive recursive.

I understand that the ackermann function terminates, and thus is a total function. So why is it not primitive recursive? The only information that i can find on the wikipedia page is

[Ackermann's function] grows faster than any primitive recursive function and is therefore not primitive recursive

Which isn't a good example of why it is not primitive recursive.

Could anyone explain to me how exactly ackermann's function is NOT primitive recursive please?

You seem to be conflating primitive recursive functions with total recursive functions. Actually, primitive recursive are a subset of total recursive functions. The hierarchy can be described informally as follows:

• Partial recursive functions are defined by an algorithm which only works on some inputs.
• Total recursive functions are defined by an algorithm which works on all inputs but can take an arbitrarily long time to compute.
• Primitive recursive functions are defined by an algorithm that completes in a knowable time (“not too long” for some theoretic definition of long).

More precisely (though I'll refer you to Wikipedia or some other reference for a complete definition), a primitive recursive function can be computed by a sequence of elementary computations where there is a way to bound the running time at every step. The only form of recursion allowed in the definition is primitive recursion, where a function calls itself on a smaller argument. All primitive recursions can be reduced to giving definitions for $f(0)$ (not involving $f$) and $f(n+1)$ as a function of $f(n)$ only.

General recursive definitions allow an extra operation: looking for the smallest $n$ that makes some property come true. In general, it's impossible to know how far you'll have to go to find such an $n$.

The definition of the Ackermann function contains the clause $A(m+1,n+1) = A(m, A(m+1, n))$. The “next” value of the function (going from $n$ to $n+1$) depends on calls with larger parameters (starting with $A(m+1,n)$). So the definition is not in primitive recursive form. It takes some work which I am not going to reproduce here, but you can show that there is no alternate presentation of the Ackermann function that uses only primitive recursion. Yet the function is well-defined, because computing $A(?,n)$ only requires computations of $A(?,n')$ with $n' < n$.

If you prefer a programming approach, recursive functions can be computed with dynamic memory allocation and while loops. Primitive recursive functions can be computed with only for loops, where the loop bound may be computed at run time but must be known when the loop starts.

Here's a proof showing why Ackermann's function is not primitive recursive.

The key to showing that A is not primitive recursive, is to find a properties shared by all primitive recursive functions, but not by A . One such property is in showing that A in some way ``grows'' faster than any primitive recursive function

Also, here's a proof showing that Ackermann's function is both a total function and a recursive function.