Prove that the equation has no integer solutions

Question

Prove that for infinitely many integers $n>2$ equation
$a^n-(a-2)^n=b^{n-1}$
has no integer solutions for $a,b$. Edit: I would appreciate any hints. They may concern other nonlinear diophantine equations for n>... as I think I wouldn't be able to do even another example and I guess I may not be the only one having these difficulties

Answer 1

The equation doesn't have integer solutions, if $n$ is an odd prime: because of the theorem of Fermat, we have $x^{n-1}=1\pmod n$, for $x\neq0\pmod n$, so that always $x^n=x\pmod n$. That means $a^n-(a-2)^n=a-(a-2)=2\pmod n$. But $b^{n-1}\pmod n$ is $0$ if $n$ divides $b$, and $1$ otherwise, so the equation can't be satisfied.