Subfield: Number Theory


Question: Find all surjective functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for every $m, n \in \mathbb{N}$ and every prime $p$, the number $f(m+n)$ is divisible by $p$ if and only if $f(m)+f(n)$ is divisible by $p$.

( $\mathbb{N}$ is the set of all positive integers.)

Solution:
Suppose that function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies the problem conditions.

Lemma. For any prime $p$ and any $x, y \in \mathbb{N}$, we have $x \equiv y(\bmod p)$ if and only if $f(x) \equiv f(y)$ $(\bmod p)$. Moreover, $p \mid f(x)$ if and only if $p \mid x$.

Proof. Consider an arbitrary prime $p$. Since $f$ is surjective, there exists some $x \in \mathbb{N}$ such that $p \mid f(x)$. Let
$$
d=\min \{x \in \mathbb{N}: p \mid f(x)\}
$$
By induction on $k$, we obtain that $p \mid f(k d)$ for all $k \in \mathbb{N}$. The base is true since $p \mid f(d)$. Moreover, if $p \mid f(k d)$ and $p \mid f(d)$ then, by the problem condition, $p \mid f(k d+d)=f((k+1) d)$ as required.

Suppose that there exists an $x \in \mathbb{N}$ such that $d \not x$ but $p \mid f(x)$. Let
$$
y=\min \{x \in \mathbb{N}: d \nmid x, p \mid f(x)\} .
$$
By the choice of $d$, we have $y>d$, and $y-d$ is a positive integer not divisible by $d$. Then $p \nmid f(y-d)$, while $p \mid f(d)$ and $p \mid f(d+(y-d))=f(y)$. This contradicts the problem condition. Hence, there is no such $x$, and
$$
p|f(x) \Longleftrightarrow d| x .\tag{1}
$$
Take arbitrary $x, y \in \mathbb{N}$ such that $x \equiv y(\bmod d)$. We have $p \mid f(x+(2 x d-x))=f(2 x d)$; moreover, since $d \mid 2 x d+(y-x)=y+(2 x d-x)$, we get $p \mid f(y+(2 x d-x))$. Then by the problem condition $p|f(x)+f(2 x d-x), p| f(y)+f(2 x d-x)$, and hence $f(x) \equiv-f(2 x d-x) \equiv f(y)$ $(\bmod p)$.

On the other hand, assume that $f(x) \equiv f(y)(\bmod p)$. Again we have $p \mid f(x)+f(2 x d-x)$ which by our assumption implies that $p \mid f(x)+f(2 x d-x)+(f(y)-f(x))=f(y)+f(2 x d-x)$. Hence by the problem condition $p \mid f(y+(2 x d-x))$. Using (1) we get $0 \equiv y+(2 x d-x) \equiv y-x$ $(\bmod d)$.

Thus, we have proved that
$$
x \equiv y \quad(\bmod d) \Longleftrightarrow f(x) \equiv f(y) \quad(\bmod p)\tag{2}
$$
We are left to show that $p=d$ : in this case (1) and (2) provide the desired statements.

The numbers $1,2, \ldots, d$ have distinct residues modulo $d$. By (2), numbers $f(1), f(2), \ldots$, $f(d)$ have distinct residues modulo $p$; hence there are at least $d$ distinct residues, and $p \geq d$. On the other hand, by the surjectivity of $f$, there exist $x_{1}, \ldots, x_{p} \in \mathbb{N}$ such that $f\left(x_{i}\right)=i$ for any $i=1,2, \ldots, p$. By (2), all these $x_{i}$ 's have distinct residues modulo $d$. For the same reasons, $d \geq p$. Hence, $d=p$.

Now we prove that $f(n)=n$ by induction on $n$. If $n=1$ then, by the Lemma, $p \nmid f(1)$ for any prime $p$, so $f(1)=1$, and the base is established. Suppose that $n>1$ and denote $k=f(n)$. Note that there exists a prime $q \mid n$, so by the Lemma $q \mid k$ and $k>1$.

If $k>n$ then $k-n+1>1$, and there exists a prime $p \mid k-n+1$; we have $k \equiv n-1$ $(\bmod p)$. By the induction hypothesis we have $f(n-1)=n-1 \equiv k=f(n)(\bmod p)$. Now, by the Lemma we obtain $n-1 \equiv n(\bmod p)$ which cannot be true.



Analogously, if $k<n$, then $f(k-1)=k-1$ by induction hypothesis. Moreover, $n-k+1>1$, so there exists a prime $p \mid n-k+1$ and $n \equiv k-1(\bmod p)$. By the Lemma again, $k=f(n) \equiv$ $f(k-1)=k-1(\bmod p)$, which is also false. The only remaining case is $k=n$, so $f(n)=n$.

Finally, the function $f(n)=n$ obviously satisfies the condition.

Final answer: $f(n)=n$