This lets us reduce, in principle, all problems involving primitive recursive functions and relations to problems involving only $1$-place primitive recursive functions and relations. \begin{thm} \label{r:cd} A $k$-place $g$ is primitive recursive if and only if the $1$-place function $h$ given by $h(n) = g(n_1, \dots, n_k)$ if $n = p_1^{n_1}\dots p_k^{n_k}$ is primitive recursive. \end{thm} \begin{note} It doesn't matter what the function $h$ may do on an $n$ which does not represent a sequence of length $k$. \end{note} \begin{cor} \label{c:thirteen10} A $k$-place relation $P$ is primitive recursive if and only if the $1$-place relation $P'$ is primitive recursive, where $$ (n_1, \dots, n_k) \in P \iff p_1^{n_1}\dots p_k^{n_k} \in P' \,. $$ \end{cor} \subsection*{A computable but not primitive recursive function} While primitive recursion and composition do not quite suffice to build all Turing computable functions from the initial functions, they are powerful enough that specific counterexamples are not all that easy to find. \begin{exmp}[Ackerman's Function] \index{Ackerman's Function} \index{$\mathsc{A}$} \index{$\alpha$} \label{pr:ack} Define the $2$-place function $\mathsc{A}$ from as follows: \begin{itemize} \item $\mathsc{A}(0,\ell) = \mathsc{S}(\ell)$ \item $\mathsc{A}(\mathsc{S}(k),0) = \mathsc{A}(k,1)$ \item $\mathsc{A}(\mathsc{S}(k),\mathsc{S}(\ell)) = \mathsc{A}(k,\mathsc{A}(\mathsc{S}(k),\ell))$ \end{itemize} Given $\mathsc{A}$, define the $1$-place function $\alpha$ by $\alpha(n) = \mathsc{A}(n,n)$.

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