$a,b,m,n$ be positive integers such that g.c.d.$(a,b)=1$ and $a^m+b^m\mid a^n+b^n$ , then $m\mid n$?

Question

Let $a,b,m,n$ be positive integers such that g.c.d.$(a,b)=1$ and $a^m+b^m\mid a^n+b^n$, then is it true that $m$ divides $n$?

Answer 1

Let $n\equiv s\pmod m$, with $0\le s<m$, and $n=mk + s$. We have that

$$a^n + b^n \equiv b^{mk}((-1)^ka^s + b^s)\equiv 0 \pmod {a^m + b^m}$$ but $(b^{mk},a^m+b^m)=1$, so $$(-1)^ka^s + b^s\equiv 0 \pmod {a^m + b^m}$$ but $$|(-1)^ka^s + b^s|\le a^s + b^s < a^m+b^m $$ so $(-1)^ka^s + b^s=0$. $a,b$ are coprime, so we have necessarily $$a^s=b^s=1\implies s=0\implies m|n$$

Edit:

There is only one exception: if $a=b=1$ then any $m,n$ are ok, so the statement is true if $ab>1$.