# Section 3.3: Problem 9 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that there is a representable function $f$ such that for every $n$ , $a_{0},\ldots,a_{n}$ , (For example, $f(72)=1$ and $f(750)=2$ .)
$f(b)$ is the least $n$ such that either $b=0$ , or $b$ is odd, or $R$ : $p_{\mbox{ln}b-1}^{n+2}$ does not divide $b$ . Here, $\in R$ iff for all $l and $p\le b$ , if $l=\mbox{lh}b-1$ and $p=p_{l}$ , then $\notin\mathcal{D}$ , where $\mathcal{D}$ is the divisibility relation. This works, because $\mbox{ln}b=0$ iff $b=0$ or $b$ is odd (in which case the function is undefined, so we just return $0$ ), otherwise, $b\ge2$ is even, $\mbox{ln}b\ge1$ , and $p_{\mbox{ln}b-1}^{n+2}$ does not divide $b$ for some large $n$ , moreover, for $b=\langle a_{0},\ldots,a_{m}\rangle$ , $\mbox{ln}b=m+1$ , and $p_{m}^{k}$ divides $b$ iff $k\le a_{m}+1$ , i.e. $k=a_{m}+2$ is the least power such that $p_{m}^{k}$ does not divide $b$ .