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.
@article{119363,
author = {Amir Khosravi and Behrooz Khosravi},
title = {On the Diophantine equation $\frac{q^n-1}{q-1}=y$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {1-7},
zbl = {1097.11015},
mrnumber = {2045841},
language = {en},
url = {http://dml.mathdoc.fr/item/119363}
}
Khosravi, Amir; Khosravi, Behrooz. On the Diophantine equation $\frac{q^n-1}{q-1}=y$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 1-7. http://gdmltest.u-ga.fr/item/119363/
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