Abstract:There exist many results about the Diophantine equation $(q^n-1)/(q-1)=y^m$, where $m\ge 2$ and $n\geq 3$. In this paper, we suppose that $m=1$, $n$ is an odd integer and $q$ a power of a prime number. Also let $y$ be an integer such that the number of prime divisors of $y-1$ is less than or equal to $3$. Then we solve completely the Diophantine equation $(q^n-1)/(q-1)=y$ for infinitely many values of $y$. This result finds frequent applications in the theory of finite groups.
Keywords: higher order Diophantine equation, exponential Diophantine equation
AMS Subject Classification: 11D61, 11D41