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

Amir Khosravi; Behrooz Khosravi — 2003

Commentationes Mathematicae Universitatis Carolinae

There exist many results about the Diophantine equation $(q^{n} - 1) / (q - 1) = y^{m}$ , where $m \geq 2$ 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.

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

A new characterization of some alternating and symmetric groups.

2-recognizability by prime graph of $PSL (2, p^{2})$ .

Quasirecognition by prime graph of the simple group $^{2} G_{2} (q)$ .

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Currently displaying 1 – 4 of 4

On the Diophantine equation q n - 1 q - 1 = y

A new characterization of some alternating and symmetric groups.

2-recognizability by prime graph of PSL ( 2 , p 2 ) .

Quasirecognition by prime graph of the simple group 2 G 2 ( q ) .

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

2-recognizability by prime graph of $PSL (2, p^{2})$ .

Quasirecognition by prime graph of the simple group $^{2} G_{2} (q)$ .