- Thread starter
- #1

- Apr 13, 2013

- 3,828

How can I prove that there are no $a,b \geq 1,n \geq 2$ such that: $a^n-b^n|a^n+b^n$ ?

I have tried the following..Could you tell me if it is right so far and how I could continue?

We suppose that $(a,b)=d$.We know that $(\frac{a}{d},\frac{b}{d})=1$.

$d|a,d|b \Rightarrow a=k \cdot d,b=l \cdot d ,k,l \in \mathbb{Z}$

Then we suppose that $a^n-b^n|a^n+b^n$.Then, $\frac{a^n+b^n}{a^n-b^n} \in \mathbb{Z}$.

$\frac{a^n+b^n}{a^n-b^n}=1+\frac{2l^n}{k^n-l^n}$