Subfield: Number Theory


Question: For every $n \in \mathbb{N}$ let $d(n)$ denote the number of (positive) divisors of $n$. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ with the following properties:

(i) $d(f(x))=x$ for all $x \in \mathbb{N}$;

(ii) $f(x y)$ divides $(x-1) y^{x y-1} f(x)$ for all $x, y \in \mathbb{N}$.

Solution:
There is a unique solution: the function $f: \mathbb{N} \rightarrow \mathbb{N}$ defined by $f(1)=1$ and 
$$
f(n)=p_{1}^{p_{1}^{a_{1}}-1} p_{2}^{p_{2}^{a_{2}}-1} \cdots p_{k}^{p_{k}^{a_{k}}-1} \text{where }n=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{k}^{a_{k}} \text{ is the prime factorization of } n>1 \tag{1}
$$

Direct verification shows that this function meets the requirements.

Conversely, let $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfy (i) and (ii). Applying (i) for $x=1$ gives $d(f(1))=1$, so $f(1)=1$. In the sequel we prove that (1) holds for all $n>1$. Notice that $f(m)=f(n)$ implies $m=n$ in view of (i). The formula $d\left(p_{1}^{b_{1}} \cdots p_{k}^{b_{k}}\right)=\left(b_{1}+1\right) \cdots\left(b_{k}+1\right)$ will be used throughout.

Let $p$ be a prime. Since $d(f(p))=p$, the formula just mentioned yields $f(p)=q^{p-1}$ for some prime $q$; in particular $f(2)=q^{2-1}=q$ is a prime. We prove that $f(p)=p^{p-1}$ for all primes $p$.

Suppose that $p$ is odd and $f(p)=q^{p-1}$ for a prime $q$. Applying (ii) first with $x=2$, $y=p$ and then with $x=p, y=2$ shows that $f(2 p)$ divides both $(2-1) p^{2 p-1} f(2)=p^{2 p-1} f(2)$ and $(p-1) 2^{2 p-1} f(p)=(p-1) 2^{2 p-1} q^{p-1}$. If $q \neq p$ then the odd prime $p$ does not divide $(p-1) 2^{2 p-1} q^{p-1}$, hence the greatest common divisor of $p^{2 p-1} f(2)$ and $(p-1) 2^{2 p-1} q^{p-1}$ is a divisor of $f(2)$. Thus $f(2 p)$ divides $f(2)$ which is a prime. As $f(2 p)>1$, we obtain $f(2 p)=f(2)$ which is impossible. So $q=p$, i. e. $f(p)=p^{p-1}$.

For $p=2$ the same argument with $x=2, y=3$ and $x=3, y=2$ shows that $f(6)$ divides both $3^{5} f(2)$ and $2^{6} f(3)=2^{6} 3^{2}$. If the prime $f(2)$ is odd then $f(6)$ divides $3^{2}=9$, so $f(6) \in\{1,3,9\}$. However then $6=d(f(6)) \in\{d(1), d(3), d(9)\}=\{1,2,3\}$ which is false. In conclusion $f(2)=2$.

Next, for each $n>1$ the prime divisors of $f(n)$ are among the ones of $n$. Indeed, let $p$ be the least prime divisor of $n$. Apply (ii) with $x=p$ and $y=n / p$ to obtain that $f(n)$ divides $(p-1) y^{n-1} f(p)=(p-1) y^{n-1} p^{p-1}$. Write $f(n)=\ell P$ where $\ell$ is coprime to $n$ and $P$ is a product of primes dividing $n$. Since $\ell$ divides $(p-1) y^{n-1} p^{p-1}$ and is coprime to $y^{n-1} p^{p-1}$, it divides $p-1$; hence $d(\ell) \leq \ell<p$. But (i) gives $n=d(f(n))=d(\ell P)$, and $d(\ell P)=d(\ell) d(P)$ as $\ell$ and $P$ are coprime. Therefore $d(\ell)$ is a divisor of $n$ less than $p$, meaning that $\ell=1$ and proving the claim.

Now (1) is immediate for prime powers. If $p$ is a prime and $a \geq 1$, by the above the only prime factor of $f\left(p^{a}\right)$ is $p$ (a prime factor does exist as $f\left(p^{a}\right)>1$ ). So $f\left(p^{a}\right)=p^{b}$ for some $b \geq 1$, and (i) yields $p^{a}=d\left(f\left(p^{a}\right)\right)=d\left(p^{b}\right)=b+1$. Hence $f\left(p^{a}\right)=p^{p^{a}-1}$, as needed.

Let us finally show that (1) is true for a general $n>1$ with prime factorization $n=p_{1}^{a_{1}} \cdots p_{k}^{a_{k}}$. We saw that the prime factorization of $f(n)$ has the form $f(n)=p_{1}^{b_{1}} \cdots p_{k}^{b_{k}}$. For $i=1, \ldots, k$, set $x=p_{i}^{a_{i}}$ and $y=n / x$ in (ii) to infer that $f(n)$ divides $\left(p_{i}^{a_{i}}-1\right) y^{n-1} f\left(p_{i}^{a_{i}}\right)$. Hence $p_{i}^{b_{i}}$ divides $\left(p_{i}^{a_{i}}-1\right) y^{n-1} f\left(p_{i}^{a_{i}}\right)$, and because $p_{i}^{b_{i}}$ is coprime to $\left(p_{i}^{a_{i}}-1\right) y^{n-1}$, it follows that $p_{i}^{b_{i}}$ divides $f\left(p_{i}^{a_{i}}\right)=p_{i}^{p_{i}^{a_{i}}-1}$. So $b_{i} \leq p_{i}^{a_{i}}-1$ for all $i=1, \ldots, k$. Combined with (i), these conclusions imply

$$
p_{1}^{a_{1}} \cdots p_{k}^{a_{k}}=n=d(f(n))=d\left(p_{1}^{b_{1}} \cdots p_{k}^{b_{k}}\right)=\left(b_{1}+1\right) \cdots\left(b_{k}+1\right) \leq p_{1}^{a_{1}} \cdots p_{k}^{a_{k}} .
$$

Hence all inequalities $b_{i} \leq p_{i}^{a_{i}}-1$ must be equalities, $i=1, \ldots, k$, implying that (1) holds true. The proof is complete.

Final answer: the function $f: \mathbb{N} \rightarrow \mathbb{N}$ defined by $f(1)=1$ and 
$$
f(n)=p_{1}^{p_{1}^{a_{1}}-1} p_{2}^{p_{2}^{a_{2}}-1} \cdots p_{k}^{p_{k}^{a_{k}}-1} \text{where }n=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{k}^{a_{k}} \text{ is the prime factorization of } n>1
$$