## Amir Khosravi, Behrooz Khosravi

*On the Diophantine equation $\frac {q^n-1}{q-1}=y$ *

Comment.Math.Univ.Carolinae 44,1 (2003) 1-7. **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

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