Geodesic
On the Diophantine equation $\frac{q^n-1}{q-1}=y$
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 1, pp. 1-7

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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.
Classification : 11D41, 11D61
Keywords: higher order Diophantine equation; exponential Diophantine equation 
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Khosravi, Amir; Khosravi, Behrooz. On the Diophantine equation $\frac{q^n-1}{q-1}=y$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 1, pp. 1-7. http://geodesic.mathdoc.fr/item/CMUC_2003__44_1_a0/