Solving Diophatine Equations

Question

Find all positive integers $a$ and $b$ such that $a^n+1=b^{n+1}$ and gcd$(n+1, a)=1$.

This is a problem from 1998 Indian olympiad (though I believe it is originally from Bulgaria or East Europe). It admits a purely elementary solution.

Answer 1

This is probably overkill, but... Mihăilescu's theorem, better known as Catalan's conjecture, says that the only two consecutive perfect powers are $8$ and $9$, but $a^n=8$ and $b^{n+1}=9$ don't give integer solutions, so $n>1$ is not possible. Then $n=1$, so $a+1=b^2$ and $a$ is odd, which has infinitely many solutions for all even $b$.