Suppose $n$ divides $3^n + 4^n$. Show that $7$ divides $n$.

Question

Suppose $n \geq 2$ and $n$ is a divisor of $3^n + 4^n$. Prove that $7$ is a divisor of $n$.

My work so far:

I had a hypothesis that if $n| 3^n + 4^n$, then $n = 7^k$ for some $k\in\mathbb{N}$. But this is not necessarily so. Take $n = 7⋅379$, where $3^7 + 4^7 = 7^2⋅379$. Then, $3^7+4^7$ divides $3^n + 4^n$, and since $n$ divides $3^7+4^7$, we must have $n|3^n+4^n$.

Answer 1

First, we note that $3 \nmid 3^n + 4^n$ and $4 \nmid 3^n + 4^n$. So $3 \nmid n$ and $2 \nmid n$.

Now, reworking $3^n + 4^n \equiv 0 \pmod n$. Since n is odd, we find \begin{equation} 3^n \equiv (-4)^n \pmod n. \end{equation} Since $\gcd(3, n) = 1$, we can take the inverse of $3 \mod n$, i. e. there exists $3^{-1}$ so that $3 \cdot 3^{-1} \equiv 1 \pmod n$. Multiplying both sides by $(3^{-1})^n$ gives \begin{equation} (-4 \cdot 3^{-1})^n \equiv 1 \pmod n. \end{equation}

Now, if $O_m(k)$ denotes the order of $k \mod m$, we know from algebra that \begin{equation} O(-4 \cdot 3^{-1}) \mid n. \end{equation}

Now, to show that $7 \nmid n$ for any solution, let's assume $7 \nmid n$ for some $n \in \mathbb{N}$. Let's assume $n > 1$ is the smallest solution so that $7 \nmid n$ and $n \mid 3^n + 4^n$. If we can show that either $n = 1$ or $7 \mid n$ or there exists an $m < n$ satisfying these properties, we're done.

Let's split 2 cases:

(Case 1): $O_n(-4 \cdot 3^{-1}) = 1$ Then we're done, since that implies (using uniqueness of inverses): \begin{equation} (-4 \cdot 3^{-1}) \equiv 1 \pmod n \implies -4 \cdot 3^{-1} \equiv 3 \cdot 3^{-1} \pmod n \\ \implies -4 \equiv 3 \pmod n \implies n \mid 7 \implies n = 1 \vee n = 7. \end{equation}

(Case 2): $O_n(-4 \cdot 3^{-1}) > 1$. Let $d = O_n(-4 \cdot 3^{-1})$. Then $(-4 \cdot 3^{-1})^d \equiv 1 \pmod n$, and since $d \mid n$ we have $(-4 \cdot 3^{-1})^d \equiv 1 \pmod d$. $d$ is also odd, and \begin{equation} -4^d (3^{-1})^d \equiv 1 \pmod d \implies -4^d \equiv 3^d \pmod d \end{equation}

If $O_d(-4 \cdot 3^{-1}) = 1$, then by the reasoning above $d \mid 7$. Since $d > 1$, we have $d = 7$. Since $d \mid n$, $7 \mid n$.

If $O_d(-4 \cdot 3^{-1}) > 1$, let m = $O_d(-4 \cdot 3^{-1})$. Then $m > 1$, $7 \nmid m$ and $m \mid 3^m + 4^m$. $m < n$, since $m \mid \varphi(n)$. So we found an $m < n$ satisfying the above conditions, implying $n$ is not the smallest.

Answer 2

Assume $\enspace n\mid 3^n+4^n$, for some $n\in\mathbb{N}$, $n\geq 2$.

Hence, $$3^n+4^n=nm, \quad for \enspace m\in\mathbb{N}.$$

Since $3^n+4^n$ is odd $\enspace (odd\cdot odd=odd, \enspace even\cdot even=even, \enspace odd+even=odd)$, and since $\enspace n \mid 3^n+4^n$, we know that $n$ is also odd. Thus, we can express $3^n+4^n$ as $$3^n+4^n=(3+4)\bigg(3^{n-1}-3^{n-2}4+3^{n-3}4^2-\ldots +3^24^{n-3}-34^{n-2}+4^{n-1}\bigg)$$

$$3^n+4^n=7k \qquad$$ whereby $\enspace k=\big(3^{n-1}-3^{n-2}4+3^{n-3}4^2-\ldots +3^24^{n-3}-34^{n-2}+4^{n-1}\big).$

Since $\enspace 7 \mid 7k$, it follows that $\enspace 7 \mid 3^n+4^n$.

Thus, so far we have, $$3^n+4^n=nm=7k$$

and hence, $$n=7\bigg(\frac{k}{m} \bigg)$$

If $\enspace m \mid k$, then great, we are done!

So, let's assume the contrary that $\enspace m \nmid k$. Since $\enspace m \nmid k$, there exists $q\in\mathbb{N}, q>0$ such that $m=7q$ and $n=\frac{k}{q}$, whereby $q\mid k$. $\enspace 7\nmid n$, since $\frac{n}{7}=\frac{k}{m}$. Hence, $gcd(n,7)=1$. We have already shown that $7\mid 7k$ and $n\mid 7k$. Thus,

$$gcd(n,7)=1, \enspace 7\mid 7k, \enspace n\mid 7k \enspace \Longrightarrow \enspace 7n\mid 7k \enspace \Longrightarrow \enspace n\mid k \enspace \Longrightarrow \enspace \frac{k}{q}\mid k$$

Since $m\nmid k$, $k=mb+r$ for some $b,r\in \mathbb{Z}, \enspace 0\leq r < m$. For some $s\in\mathbb{N}$,

$$\frac{k}{q}\mid k \enspace \Longrightarrow \enspace \frac{mb+r}{q}\mid k \enspace \Longrightarrow \enspace \big(7b+\frac{r}{q}\big)\mid k \enspace \Longrightarrow \enspace k=s\big(7b+\frac{r}{q}\big)$$

$$k=7sb+\frac{rs}{q}=mb+r \enspace \Longrightarrow \enspace b(7s-m)=r\big(1-\frac{s}{q}\big) \enspace \Longrightarrow \enspace \frac{7bq}{r}=\frac{q-s}{s-q} \enspace \Longrightarrow \enspace \frac{7bq}{r}=-1 \enspace \Longrightarrow \enspace 7bq=-r \enspace \Longrightarrow \enspace r=-mb \enspace \Longrightarrow \enspace k=mb-mb=0$$

But we already know that $k>0$, hence a contradiction. Thus, our assumption is false, and $m\mid k$.

Hence, since $n=7\big(\frac{k}{m}\big)$ and $m \mid k$, therefore $7\mid n$.