Complete $\mu$-recursive function that is not primitive recursive

Question

Could you find an example of a complete $\mu$-recursive function that is not a primitive function?

Answer 1

There are two standard approaches: diagonalization and rate of growth.

The definition of primitive recursive functions (whichever one you use) allows enumerating them $f_1,f_2,\ldots$. Since each primitive recursive function is total, the function $f(n) = f_n(n)+1$ is recursive. It is clearly not primitive recursive.

The other approach is exemplified by the Ackermann function, which is clearly recursive but grows too fast to be primitive recursive.