number theory - To prove that $n$ does not divide $1+(3^n)$.

$n$ is an odd integer greater than $1$. How do I prove that $n$ does not divide $1+(3^n)$. I have proved the case when $n$ is a prime by using Fermat's Theorem. Can I get some help for the general case?

Answer

Prove it by contradiction.

The case $n=3$ is obvious. Let $n\ge 5$.

Let $p$ be the least prime divisor of $n$ ($p$ must be odd and greater than $3$).

Then $3^{2n}\equiv 1\pmod{p}$ and $3^{p-1}\equiv 1\pmod{p}$

by Fermat's Little Theorem.

I.e. $\text{ord}_p(3)\mid 2n$ and $\text{ord}_p(3)\mid p-1$.

I.e. $\text{ord}_p(3)\mid \gcd(2n,p-1)=2$.

I.e. either $3^2\equiv 1\pmod{p}$ or $3\equiv 1 \pmod{p}$.

I.e. either $p\mid 8$ or $p\mid 2$. But $p$ must be odd, contradiction.