Does there exist positive integers $a,b,n$ , where $n>1$ , such that $a^n - b^n |a^n + b^ n$ ?

Question

Does there exist positive integers $a,b,n$ , where $n>1$ , such that $a^n - b^n |a^n + b^ n$ ? ; the only trivial thing I can see is that if so happens , then $a^n - b^n | 2b^n$ , but nothing else . Please help . Thanks in advance

Answer 1

If there exist such positive $a$ and $b$, then there are relatively prime $a$ and $b$ such that $a^n-b^n$ divides $a^n+b^n$.

Let $p$ be an odd prime that divides $a^n-b^n$. Then $p$ divides $a^n+b^n$. It follows by your observation that $p$ divides $2b^n$ and $2a^n$, contradicting relative primality.

So the only possibility left with $a$ and $b$ relatively prime is if $a^n-b^n$ is a power of $2$. Since $a^n-b^n$ divides $2a^n$ and $2b^n$, it follows that we must have $a^n-b^n=2$, which cannot happen if $n\ge 2$.